Unformatted text preview: 1 Numerical Methods in Fluid Dynamics
MPO 662
Instructor Mohamed Iskandarani
MSC 320 x 4045
[email protected]
Grades 60% Homework (involve programming)
20% Mid term
20% Term project Syllabus
1. Introduction
2. Classiﬁcations of PDE’s and their properties
3. Basics of the ﬁnite diﬀerence method
4. Finite diﬀerence solutions of ODE
5. Finite diﬀerence solutions of timedependent linear PDEs
(a) advection equation
(b) heat equation
(c) Stability and dispersion properties of time diﬀerencing schemes
6. Numerical solution of ﬁnite diﬀerence approximation of elliptic equations
7. Special advection schemes
8. Energetically consistant ﬁnite diﬀerence schemes
9. The Finite Element Method
10. Additional topics (time permitting) 2 Background 1. Name 2. Degree Sought (what ﬁeld) 3. Advisor (if any) 4. Background 5. Scientiﬁc Interest 6. Background in numerical modeling 7. Programming experience/language 3 Reserve List
• Dale B. Haidvogel and Aike Beckmann, Numerical Ocean Circulation Modeling
Imperial College Press, 1999. (CGFD)
• Dale R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer, New York, 1998. (CGFD)
• George J. Haltiner Roger T. Williams, Numerical Prediction and Dynamic Meteorology, Wiley, 1980. (CGFD)
• John C. Tannehill, Dale A. Anderson, and Richard H. Pletcher, Computational
Fluid Mechanics and Heat Transfer, Taylor and Francis, 1997. (FDM)
• G. E. Forsythe and W. R. Wasow FiniteDiﬀerence Methods for Partial Diﬀerential
Equations, John Wiley and Sons, Inc., New York, 1960. (FDM)
• R. D. Richtmyer and K. W. Morton, Diﬀerence Methods for Initial–Value Problems,
Interscience Publishers (J. Wiley & Sons), New York, 1967. Useful References
• Gordon D. Smith, Numerical Solution of Partial Diﬀerential Equations : Finite
Diﬀerence Methods, Oxford University Press, New York, 1985. (FDM)
• K.W. Morton and D.F. Mayers, Numerical Solution of Partial Diﬀerential Equations : An Introduction, Cambridge University Press, New York, 1994. (FDM)
• P.J. Roache, Computational Fluid Dynamics, Hermosa Publisher, 1972, ISBN 0913478059. (FDM)
• C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, 2 volumes, 2nd
ed., SpringerVerlag, New York, 19911992. (Num. Sol. of PDE’s)
• Roger Peyret and Thomas D. Taylor, Computational Methods for Fluid Flow,
SpringerVerlag, New York, 1990. (Num. Sol. of PDE’s)
• Roger Peyret, Handbook of Computational Fluid Mechanics, Academic Press, San
Diego, 1996. (QA911 .H347 1996)
• Joel H. Ferziger and M. Peric Computational Methods For Fluid Dynamics, SpringerVerlag, New York, 1996.
• R. S. Varga, Matrix Iterative Analysis, Prentice–Hall, New York, 1962.
• Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1998. (Spectral methods)
• C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid
Dynamics, SpringerVerlag, New York, 1991. (Spectral Methods) 4
• John P. Boyd, Chebyshev and Fourier Spectral Methods Dover Publications, 2000.
(Spectral methods)
• O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, 4th edition, Mc
Graw Hill, 1989.
• George Em. Karniadakis and Spencer J. Sherwin, Spectral h − p Element Methods
for CFD, New York, Oxford University Press, 1999. (Spectral Elements)
• Michel O. Deville, Paul F. Fischer and E.H. Mund, HighOrder Methods for Incompressible Fluid Flow , Cambridge Monographs on Applied and Computational
Mathematics, Cambridge University Press, Cambridge, 2002. Useful Software
• Plotting Software (e.g. matlab, NCAR Graphics, gnuplot)
• Linear Algebra (e.g. matlab, LAPACK, IMSL)
• Fast Fourier Transforms (e.g. matlab, ﬀtpack, ?)
• Fortran Compiler (debuggers are useful too) Numerical Methods in Ocean Modeling
Lecture Notes for MPO662 October 26, 2011 2 Contents
1 Introduction
1.1 Justiﬁcation of CFD . . . . . . . . .
1.2 Discretization . . . . . . . . . . . . .
1.2.1 Finite Diﬀerence Method . .
1.2.2 Finite Element Method . . .
1.2.3 Spectral Methods . . . . . . .
1.2.4 Finite Volume Methods . . .
1.2.5 Computational Cost . . . . .
1.3 Initialization and Forcing . . . . . .
1.4 Turbulence . . . . . . . . . . . . . .
1.5 Examples of system of equations and .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
. 9
9
9
11
11
11
12
12
13
13
13 2 Basics of PDEs
2.1 Classiﬁcation of second order PDEs . . . . . . . . . . .
2.1.1 Hyperbolic Equation: b2 − 4ac > 0 . . . . . . .
2.1.2 Parabolic Equation: b2 − 4ac = 0 . . . . . . . .
2.1.3 Elliptic Equation: b2 − 4ac < 0 . . . . . . . . .
2.2 WellPosed Problems . . . . . . . . . . . . . . . . . . .
2.3 First Order Systems . . . . . . . . . . . . . . . . . . .
2.3.1 Scalar Equation . . . . . . . . . . . . . . . . . .
2.3.2 System of Equations in onespace dimension . .
2.3.3 System of equations in multispace dimensions .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
. 17
18
19
21
23
23
24
24
27
28 3 Finite Diﬀerence Approximation of Derivatives
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Finite Diﬀerence Approximation . . . . . . . . . . . . .
3.3 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Taylor series and ﬁnite diﬀerences . . . . . . . .
3.3.2 Higher order approximation . . . . . . . . . . . .
3.3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Systematic Derivation of higher order derivative
3.3.5 Discrete Operator . . . . . . . . . . . . . . . . .
3.4 Polynomial Fitting . . . . . . . . . . . . . . . . . . . . .
3.4.1 Linear Fit . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Quadratic Fit . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. 31
31
31
33
34
35
36
40
41
42
42
43 3 ...
...
...
...
...
...
...
...
...
their ......
......
......
......
......
......
......
......
......
properties 4 CONTENTS 3.5 3.4.3 Higher order formula . . . . . . . . . .
Compact Diﬀerencing Schemes . . . . . . . .
3.5.1 Derivation of 3term compact schemes
3.5.2 Families of Fourth order schemes . . .
3.5.3 Families of Sixth order schemes . . . .
3.5.4 Numerical experiments . . . . . . . . . 4 Application of Finite Diﬀerences to ODE
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 Forward Euler Approximation . . . . . . . .
4.3 Stability, Consistency and Convergence . .
4.3.1 Lax Richtmeyer theorem . . . . . . .
4.3.2 Von Neumann stability condition . .
4.4 Backward Diﬀerence . . . . . . . . . . . . .
4.5 Backward Diﬀerence . . . . . . . . . . . . .
4.6 Trapezoidal Scheme . . . . . . . . . . . . .
4.6.1 Phase Errors . . . . . . . . . . . . .
4.7 Higher Order Methods . . . . . . . . . . . .
4.7.1 Multi Stage (Runge Kutta) Methods
4.7.2 Remarks on RK schemes . . . . . . .
4.7.3 Multi Time Levels Methods . . . . .
4.8 Strongly Stable Schemes . . . . . . . . . . .
4.8.1 Stability of BDF . . . . . . . . . . .
4.9 Systems of ODEs . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 5 Numerical Solution of PDE’s
5.1 Introduction . . . . . . . . . . . . . . . . . . .
5.1.1 Convergence . . . . . . . . . . . . . .
5.1.2 Truncation Error . . . . . . . . . . . .
5.1.3 Consistency . . . . . . . . . . . . . . .
5.1.4 Stability . . . . . . . . . . . . . . . . .
5.1.5 LaxRichtmeyer Equivalence theorem
5.2 Truncation Error . . . . . . . . . . . . . . . .
5.3 The Lax Richtmeyer theorem . . . . . . . . .
5.4 The Von Neumann Stability Condition . . . .
5.5 Von Neumann Stability Analysis . . . . . . .
5.6 Modiﬁed Equation . . . . . . . . . . . . . . . .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. 6 Numerical Solution of the Advection Equation
6.1 Introduction . . . . . . . . . . . . . . . . . . . . .
6.2 Donor Cell scheme . . . . . . . . . . . . . . . . .
6.2.1 Remarks . . . . . . . . . . . . . . . . . . .
6.3 Backward time centered space (BTCS) . . . . . .
6.3.1 Remarks . . . . . . . . . . . . . . . . . . .
6.4 Centered time centered space (CTCS) . . . . . . .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. 43
46
46
47
48
48 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 49
49
51
53
53
55
55
56
56
57
59
59
61
62
66
68
69 .
.
.
.
.
.
.
.
.
.
. 71
71
72
73
73
73
73
73
74
76
77
77 .
.
.
.
.
. 81
81
81
81
84
84
87 CONTENTS 6.5
6.6 5 6.4.1 Remarks . . . . . . . . . . . . .
Lax Wendroﬀ scheme . . . . . . . . . .
6.5.1 Remarks . . . . . . . . . . . . .
Numerical Dispersion . . . . . . . . . .
6.6.1 Analytical Dispersion Relation
6.6.2 Numerical Dispersion Relation: ............
............
............
............
............
Spatial Diﬀerencing .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. 87
88
90
90
91
91 7 Finite Volume Method
7.1 The partial diﬀerential equation . . . . . . . . . . .
7.2 Integral Form of Conservation Law . . . . . . . . .
7.3 Sketch of Finite Volume Methods . . . . . . . . . .
7.4 Finite Volume in 1D . . . . . . . . . . . . . . . . .
7.4.1 Function Reconstruction . . . . . . . . . . .
7.4.2 Piecewise constant . . . . . . . . . . . . . .
7.4.3 Piecewise Linear . . . . . . . . . . . . . . .
7.4.4 Piecewise parabolic . . . . . . . . . . . . . .
7.4.5 Reconstruction Validation . . . . . . . . . .
7.5 Finite Volume Method for Scalar Advection in 2D
7.5.1 Function reconstruction in 2D . . . . . . . .
7.6 Algorithm Summary . . . . . . . . . . . . . . . . .
7.7 Code Design . . . . . . . . . . . . . . . . . . . . . .
7.7.1 Data Structure . . . . . . . . . . . . . . . .
7.7.2 Domain Geometry . . . . . . . . . . . . . .
7.7.3 Flow . . . . . . . . . . . . . . . . . . . . . .
7.7.4 T initiations . . . . . . . . . . . . . . . . .
7.8 Tracer Advection in a Stommel Gyre . . . . . . . .
7.8.1 The ﬂow ﬁeld . . . . . . . . . . . . . . . . .
7.8.2 The initial condition . . . . . . . . . . . . .
7.8.3 Expected result . . . . . . . . . . . . . . . .
7.8.4 Support Code . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 95
95
96
98
99
99
100
100
101
102
107
110
112
113
113
113
114
114
114
114
115
117
117 8 Numerical Dispersion of Linearized
8.1 Linearized SWE in 1D . . . . . . .
8.1.1 Centered FDA on Agrid .
8.1.2 Centered FDA on Cgrid .
8.2 TwoDimensional SWE . . . . . .
8.2.1 Inertia gravity waves . . . .
8.3 Rossby waves . . . . . . . . . . . . .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. 119
119
120
121
124
124
125 9 Solving the Poisson Equations
9.1 Iterative Methods . . . . . . . . . . . . . . . . . . .
9.1.1 Jacobi method . . . . . . . . . . . . . . . .
9.1.2 GaussSeidel method . . . . . . . . . . . . .
9.1.3 Successive Over Relaxation (SOR) method
9.1.4 Iteration by Lines . . . . . . . . . . . . . . .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. 137
139
139
140
141
141 SWE
....
....
....
....
....
.... .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. 6 CONTENTS 9.2
9.3 9.1.5 Matrix Analysis .
Krylov MethodCG . . . .
Direct Methods . . . . . .
9.3.1 Periodic Problem .
9.3.2 Dirichlet Problem .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. 10 Nonlinear equations
10.1 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 1D Burger equation . . . . . . . . . . . . . . . . . .
10.3 Quadratic Conservation . . . . . . . . . . . . . . . .
10.4 Nonlinear advection equation . . . . . . . . . . . . .
10.4.1 FD Approximation of the advection term . .
10.5 Conservation in vorticity streamfunction formulation
10.6 Conservation in primitive equations . . . . . . . . . .
10.7 Conservation for divergent ﬂows . . . . . . . . . . . .
11 Special Advection Schemes
11.1 Introduction . . . . . . . . . . . . . . . . . . . . .
11.2 Monotone Schemes . . . . . . . . . . . . . . . . .
11.3 Flux Corrected Transport (FCT) . . . . . . . . .
11.3.1 OneDimensional . . . . . . . . . . . . . .
11.3.2 OneDimensional Flux Correction Limiter
11.3.3 Properties of FCT . . . . . . . . . . . . .
11.3.4 TwoDimensional FCT . . . . . . . . . . .
11.3.5 TimeDiﬀerencing with FCT . . . . . . .
11.4 Slope/Flux Limiter Methods . . . . . . . . . . .
11.5 MPDATA . . . . . . . . . . . . . . . . . . . . . .
11.6 WENO schemes in vertical . . . . . . . . . . . .
11.6.1 Function reconstruction . . . . . . . . . .
11.6.2 WENO reconstruction . . . . . . . . . . .
11.6.3 ENO and WENO numerical experiments
11.7 Utopia . . . . . . . . . . . . . . . . . . . . . . . .
11.8 LaxWendroﬀ for advection equation . . . . . . .
11.9 2D Numerical experiments . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
. 144
146
149
149
149 .
.
.
.
.
.
.
. 151
151
153
154
156
156
158
161
163 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 165
165
166
166
166
168
169
173
174
175
176
177
177
180
181
183
187
188 12 Fourier series
199
12.1 Continuous Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12.2 Discrete Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
12.2.1 Fourier Series For Periodic Problems . . . . . . . . . . . . . . . . . 201
13 Spectral Methods
13.1 Spectral Series . . . . . . . . . . . .
13.2 Fourier Series . . . . . . . . . . . . .
13.2.1 Bounds on Fourier coeﬃcients
13.3 Equal Error Assumptions . . . . . . .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
. 205
205
206
209
210 CONTENTS 7 14 Finite Element Methods
14.1 MWR . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.1 Collocation . . . . . . . . . . . . . . . . . . . . .
14.1.2 Least Square . . . . . . . . . . . . . . . . . . . .
14.1.3 Galerkin . . . . . . . . . . . . . . . . . . . . . . .
14.2 FEM example in 1D . . . . . . . . . . . . . . . . . . . .
14.2.1 Weak Form . . . . . . . . . . . . . . . . . . . . .
14.2.2 Galerkin form . . . . . . . . . . . . . . . . . . . .
14.2.3 Essential Boundary Conditions . . . . . . . . . .
14.2.4 Choice of interpolation and test functions . . . .
14.2.5 FEM solution using 2 linear elements . . . . . .
14.2.6 FEM solution using N linear elements . . . . . .
14.2.7 Local stiﬀness matrix and global assembly . . . .
14.2.8 Quadratic Interpolation . . . . . . . . . . . . . .
14.2.9 Spectral Interpolation . . . . . . . . . . . . . . .
14.2.10 Numerical Integration . . . . . . . . . . . . . . .
14.3 Mathematical Results . . . . . . . . . . . . . . . . . . .
14.3.1 Uniqueness and Existence of continuous solution
14.3.2 Uniqueness and Existence of continuous solution
14.3.3 Error estimates . . . . . . . . . . . . . . . . . . .
14.4 Two Dimensional Problems . . . . . . . . . . . . . . . .
14.4.1 Linear Triangular Elements . . . . . . . . . . . .
14.4.2 Higher order triangular elements . . . . . . . . .
14.4.3 Quadrilateral elements . . . . . . . . . . . . . . .
14.4.4 Interpolation in quadrilateral elements . . . . . .
14.4.5 Evaluation of integrals . . . . . . . . . . . . . . .
14.5 Timedependent problem in 1D: the Advection Equation
14.5.1 Numerical Example . . . . . . . . . . . . . . . .
14.6 The Discontinuous Galerkin Method (DGM) . . . . . .
14.6.1 Gaussian Hill Experiment . . . . . . . . . . . . .
14.6.2 Cone Experiment . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 211
211
212
212
213
213
213
214
214
215
216
220
222
224
226
228
231
231
232
232
233
234
237
238
240
242
244
247
251
252
253 15 Linear Analysis
15.1 Linear Vector Spaces . . . . . . . . . . . . .
15.1.1 Deﬁnition of Abstract Vector Space
15.1.2 Deﬁnition of a Norm . . . . . . . . .
15.1.3 Deﬁnition of an inner product . . . .
15.1.4 Basis . . . . . . . . . . . . . . . . . .
15.1.5 Example of a vector space . . . . . .
15.1.6 Function Space . . . . . . . . . . . .
15.1.7 Pointwise versus Global Convergence
15.2 Linear Operators . . . . . . . . . . . . . . .
15.3 Eigenvalues and Eigenvectors . . . . . . . .
15.4 SturmLiouville Theory . . . . . . . . . . .
15.5 Application to PDE . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. 257
257
257
258
258
258
260
260
263
263
264
264
266 .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. 8 CONTENTS 16 Rudiments of Linear Algebra 271 16.1 Vector Norms and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
16.1.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
16.1.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
16.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
16.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 273
16.4 Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
16.5 Eigenvalues of Tridiagonal Matrices . . . . . . . . . . . . . . . . . . . . . . 274 17 Programming Tips 277 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
17.2 Fortran Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
17.3 Debugging and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
17.3.1 Programming Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 281
17.3.2 Coding tips and compiler options . . . . . . . . . . . . . . . . . . . 282
17.3.3 Run time errors and compiler options . . . . . . . . . . . . . . . . 283
17.3.4 Some common pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . 284 18 Debuggers 287 18.1 Preparing the code for debugging . . . . . . . . . . . . . . . . . . . . . . . 287
18.2 Running the debugger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Chapter 1 Introduction
1.1 Justiﬁcation of CFD Fluid motion is governed by the NavierStokes equations, a set of coupled and nonlinear
partial diﬀerential equations derived from the basic laws of conservation of mass, momentum and energy. The unknowns are usually the velocity, the pressure and the density (for
stratiﬁed ﬂuids) and some tracers like temperature and salinity. The analytical paper
and pencil solution of these equations is practically impossible save for the simplest of
ﬂows. The simpliﬁcations can take the form of geometrical simpliﬁcation (the ﬂow is in
a rectangle or a circle), and/or physical simpliﬁcation (periodicity, homogeneous density,
linearity, etc...). Occasionally, it is possible to make headway by using asymptotic analyses technique, and there has been remarkable success in the past O(100) year, like the
development of boundary layer theory.
Scientists had to resort to laboratory experiments when theoretical analyses was
impossible. Physical complexity can be restored to the system. The answers delivered
are, however, usually qualitatively diﬀerent since dynamical and geometric similitudes
are diﬃcult to enforce simultaneously between the lab experiment and the prototype. A
prime example is the Reynolds’ number similarity which if violated can turn a turbulent
ﬂow laminar. Furthermore, the design and construction of these experiments can be
diﬃcult (and costly), particularly for stratiﬁed rotating ﬂows.
Computational ﬂuid dynamics (CFD) is an additional tool in the arsenal of scientists.
In its early days CFD was often controversial, as it involved additional approximation
to the governing equations and raised additional (legitimate) issues. Nowadays CFD is
an established discipline alongside theoretical and experimental methods. This position
is in large part due to the exponential growth of computer power which has allowed us
to tackle ever larger and more complex problems. 1.2 Discretization The central process in CFD is the process of discretization, i.e. the process of taking
diﬀerential equations with an inﬁnite number of degrees of freedom, and reducing it
to a system of ﬁnite degrees of freedom. Hence, instead of determining the solution
9 10 CHAPTER 1. INTRODUCTION Greenland Iceland Figure 1.1: Computation grid for a ﬁnite diﬀerence ocean model everywhere and for all times, we will be satisﬁed with its calculation at a ﬁnite number
of locations and at speciﬁed time intervals. The partial diﬀerential equations are then
reduced to a system of algebraic equations that can be solved on a computer.
Errors creep in during the discretization process. The nature and characteristics
of the errors must be controlled in order to ensure that 1) we are solving the correct
equations (consistency property), and 2) that the error can be decreased as we increase
the number of degrees of freedom (stability and convegence). Once these two criteria are
established, the power of computing machines can be leveraged to solve the problem in
a numerically reliable fashion.
Various discretization schemes have been developed to cope with a variety of issues.
The most notable for our purposes are: ﬁnite diﬀerence methods, ﬁnite volume methods,
ﬁnite element methods, and spectral methods. 1.2. DISCRETIZATION 1.2.1 11 Finite Diﬀerence Method Finite diﬀerence replace the inﬁnitesimal limiting process of derivative calculation
f (x + ∆x) − f (x)
∆x→ 0
∆x (1.1) f (x + ∆x) − f (x)
+ O(∆x)
∆x (1.2) f ′ (x) = lim
with a ﬁnite limiting process,i.e.
f ′ (x) ≈ The term O(∆x) gives an indication of the magnitude of the error as a function of the
mesh spacing. In this instance, the error is halfed if the grid spacing, ∆x is halved, and
we say that this is a ﬁrst order method. Most FDM used in practice are at least second
order accurate except in very special circumstances. We will concentrate mostly on
ﬁnite diﬀerence methods since they are still among the most popular numerical methods
for the solution of PDE’s because of their simplicity, eﬃciency, low computational cost,
and ease of analysis. Their major drawback is in their geometric inﬂexibility which
complicates their applications to general complex domains. These can be alleviated by
the use of either mapping techniques and/or masking to ﬁt the computational mesh to
the computational domain. 1.2.2 Finite Element Method The ﬁnite element method was designed to deal with problem with complicated computational regions. The PDE is ﬁrst recast into a variational form which essentially forces
the mean error to be small everywhere. The discretization step proceeds by dividing the
computational domain into elements of triangular or rectangular shape. The solution
within each element is interpolated with a polynomial of usually low order. Again, the
unknowns are the solution at the collocation points. The CFD community adopted the
FEM in the 1980’s when reliable methods for dealing with advection dominated problems
were devised. 1.2.3 Spectral Methods Both ﬁnite element and ﬁnite diﬀerence methods are low order methods, usually of 2nd4th order, and have local approximation property. By local we mean that a particular
collocation point is aﬀected by a limited number of points around it. In contrast, spectral
method have global approximation property. The interpolation functions, either polynomials or trigonomic functions are global in nature. Their main beneﬁts is in the rate of
convergence which depends on the smoothness of the solution (i.e. how many continuous
derivatives does it admit). For inﬁnitely smooth solution, the error decreases exponentially, i.e. faster than algebraic. Spectral methods are mostly used in the computations of
homogeneous turbulence, and require relatively simple geometries. Atmospheric model
have also adopted spectral methods because of their convergence properties and the
regular spherical shape of their computational domain. 12 CHAPTER 1. INTRODUCTION Figure 1.2: Elemental partition of the global ocean as seen from the eastern and western equatorial Paciﬁc. The inset shows the master element in the computational plane. The location of
the interpolation points is marked with a circle, and the structuredness of this grid local grid is
evident from the predictable adjacency pattern between collocation points. 1.2.4 Finite Volume Methods Finite volume methods are primarily used in aerodynamics applications where strong
shocks and discontinuities in the solution occur. Finite volume method solves an integral
form of the governing equations so that local continuity property do not have to hold. 1.2.5 Computational Cost The CPU time to solve the system of equations diﬀer substantially from method to
method. Finite diﬀerences are usually the cheapest on a per grid point basis followed
by the ﬁnite element method and spectral method. However, a per grid point basis
comparison is a little like comparing apple and oranges. Spectral methods deliver more
accuracy on a per grid point basis than either FEM or FDM. The comparison is more
meaningfull if the question is recast as ”what is the computational cost to achieve a given
error tolerance?”. The problem becomes one of deﬁning the error measure which is a
complicated task in general situations. 1.3. INITIALIZATION AND FORCING 1.3 13 Initialization and Forcing The state of a system is determined by its intial state, (conditions at t=0), the forces
acting on the system (sources and sinks of momentum, heat, buoyancy), the boundary
conditions, and the governing equations. Given this information one can in principle
integrate the equations forward to ﬁnd the state of the system at a future time. Things are
not so simple in practice. First, the initial state is seldom known accurately. In spite of
advances in measurements and instrumentations, data deﬁciencies still exist and manifest
themselves in either inadequate temporal or spatial coverage or measuring errors. The
situation is more diare in the ocean than the atmosphe because the dynamical scales
are smaller and require more measurement per unit area, and because of observational
diﬃculties. Furthermore, the ﬂuxes between the ocean and the atmosphere are not well
known and constitute another modeling impediment. The discipline of data assimilation
is devoted to the task of integrating data and model in optimal fashion. This is topic we
will not touch upon in this course. 1.4 Turbulence Most ﬂows occuring in nature are turbulent, in that they contain energy at all scales
ranging from hundred of kilometers to a few centimeters. It is obviously not possible to
model all these scales at once. It is often suﬃcient to model the ”large scale physics”
and to relegate the small unresolvable scale to a subgrid model. Subgrid models occupy
a large discipline in CFD, we will use the simplest of these models which rely on a simple
eddy diﬀusivity coeﬃcient. 1.5 Examples of system of equations and their properties The numerical solution of any partial diﬀerential equations should only proceed after
carefull consideration of the dynamical settings of the problem. A prudent modeler will
try to learn as much as he can about the system he is trying to solve, for ortherwise how
can he judge the results of his simulations? Errors due to numerical approximation are
sometimes notoriously hard to catch and diagnose, and an knoweldge of the expected
behavior of the system will go a long way in helping catch these errors. Furthermore, one
can often simplify the numerical solution process by taking advantages of special features
in the solution, such as symmetry or a reduction of the number of unknowns, to reduce
the complexity of the analytical and numerical formulations. In the following sections
we consider the case of the two dimensional incompressible Navier Stokes equations to
illustrate some of these points.
The incompressible NavierStokes equations is an example of a system of partial
diﬀerential equations governing a large class of ﬂuid ﬂow problems. We will conﬁne
ourselves to twodimensions for simplicity. The primtive form of these equations is:
v t + v · ∇v = − 1
∇p + ν ∇2 v
ρ0
∇·v =0 (momentum conservation) (1.3) (mass conservation) (1.4) 14 CHAPTER 1. INTRODUCTION supplemented with proper boundary and initial conditions. The unknowns are the two
components of the velocity v and the pressure p so that we have 3 unknowns functions
to determine. The parameters of the problem are the density and the kinematic viscosity
which are assumed constant in this example. Equation (1.3) expresses the conservation of
momentum, and equation (1.4), also referred to as the continuity equation, conservation
of mass which, for a constant density ﬂuid, amount to conservation of volume. The form
of equations (1.3)(1.4) is called primitive because it uses velocity and pressure as its
dependent variables.
In the early computer days memory was expensive, and modelers had to use their
ingenuity to reduce the model’s complexity as much as possible. Furthermore, the incompressibility constraint complicates the solution process substantially since there is
no simple evolution equation for the pressure. The streamfunction vorticity formulation
of the twodimensional Navier Stokes equations addresses both diﬃculties by enforcing
the continuity requirement implicitly, and reducing the number of unknown functions.
The streamfunction vorticity formulation introduces other complications in the solution
process that we will ignore for the time being. Alas, this is a typical occurence where a
cure to one set of concerns raises quite a few, and sometimes, irritating sideeﬀects. The
streamfunction is deﬁned as follows:
u = −ψy , v = ψx . (1.5) Any velocity derivable from such a streamfunction is that guaranteed to conserve mass
since
ux + vy = (−ψy )x + (ψx )y = −ψyx + ψxy = 0.
To simplify the equations further we introduce the vorticity ζ = vx − uy (a scalar in 2D
ﬂows), which in terms of the streamfunction is
∇2 ψ = ζ. (1.6) We can derive an evolution equation for the vorticity by taking the curl of the momentum
equation, Eq. (1.3). The ﬁnal form after using various vector identities are:
ζt
time rate of change
Ω
T
1
1 + v·ζ = ν ∇2 ζ advection
UΩ
L diﬀusion
νΩ
L2 UT
L
1 νT
L2
1
ν
=
UL
Re (1.7) Note that the equation simpliﬁes considerably since the pressure does not appear in the
equation (the curl of a gradient is always zero). Equations (1.6) and (1.7) are now a system of two equations in two unknowns: the vorticity and streamfunction. The pressure
has disappeared as an unknown, and its role has been assumed by the streamfunction.
The two physical processes governing the evolution of vorticity are advection by the ﬂow 1.5. EXAMPLES OF SYSTEM OF EQUATIONS AND THEIR PROPERTIES 15 and diﬀusion by viscous action. Equation (1.7) is an example of parabolic partial differential equation requiring initial and boundary data for its unique solution. Equation
(1.6) is an example of an elliptic partial diﬀerential equation. In this instance it is a
Poisson equation linking a given vorticity distribution to a streamfunction. Occasionally the term prognostic and diagnostic are applied to the vorticity and streamfunction,
respectively, to mean that the vorticity evolves dynamically in time to balance the conservation of momentum, while the streamfunction responds instantly to changes in vorticity
to enforce kinematic constraints. A numerical solution of this coupled set of equations
would proceed in the following manner: given an initial distribution of vorticity, the corresponding streamfunction is computed from the solution of the Poisson equations along
with the associated velocity; the vorticity equation is then integrated in time using the
previous value of the unknown ﬁelds; the new streamfunction is subsequently updated.
The process is repeated until the desired time is reached.
In order to gauge which process dominates, advection or diﬀusion, the vorticity evolution, we proceed to nondimensionalize the variables with the following, time, length
and velocity scales, T , L and U , respectively. The vorticity scale is then Ω = U/L from
the vorticity deﬁnition. The time rate of change, advection and diﬀusion then scale as
Ω/T , U Ω/L, and ν Ω/L2 as shown in the third line of equation 1.7. Line 4 shows the
relative sizes of the term after multiplying line 3 by T /Ω. If the time scale is chosen to be
the advective time scale, i.e. T = L/U , then we obtain line 5 which shows a single dimensionless parameter, the Reynolds number, controlling the evolution of ζ . When Re ≪ 1
diﬀusion dominates and the equation reduces to the so called heat equation ζ = ν ∇2 ζ . If
Re ≫ 1 advection dominates almost everywhere in the domain. Notice that dropping the
viscous term is problematic since it has the highest order derivative, and hence controls
the imposition of boundary conditions. Diﬀusion then has to become dominant near the
boundary through an increase of the vorticity gradient in a thin boundary layers where
advection and viscous action become balanced.
What are the implications for numerical solution of all the above analysis. By carefully analysing the vorticity dynamics we have shown that a low Reynolds number simulation requires attention to the viscous operator, whereas advection dominates in high
Reynolds number ﬂow. Furthermore, close attention must be paid to the boundary layers
forming near the edge of the domain. Further measures of checks on the solution can be
obtained by spatially integrating various forms of the vorticity equations to show that
energy, kinetic energy here, and enstrophy ζ 2 /2 should be conserved in the inviscid case,
Re = ∞, when the domain is closed. 16 CHAPTER 1. INTRODUCTION Chapter 2 Basics of PDEs
Partial diﬀerential equations are used to model a wide variety of physical phenomena.
As such we expect their solution methodology to depend on the physical principles used
to derive them. A number of properties can be used to distinguish the diﬀerent type of
diﬀerential equations encountered. In order to give concrete examples of the discussions
to follow we will use as an example the following partial diﬀerential equation:
auxx + buxy + cuyy + dux + euy + f = 0. (2.1) The unknown function in equation (2.1) is the function u which depends on the two independent variables x and y . The coeﬃcients of the equations a, b, . . . , f are yet undeﬁned.
The following properties of a partial diﬀerential equation are useful in determining
its character, properties, method of analysis, and numerical solution:
Order : The order of a PDE is the order of the highest occuring derivative. The order
in equation (2.1) is 2. A more detailed description of the equation would require
the speciﬁcation of the order for each independent variable. Equation 2.1 is second
order in both x and y . Most equations derived from physical principles, are usually
ﬁrst order in time, and ﬁrst or second order in space.
Linear : The PDE is linear if none of its coeﬃcients depend on the unknown function.
In our case this is tantamount to requiring that the functions a, b, . . . , f are independent of u. Linear PDEs are important since their linear combinations can be
combined to form yet another solution. More mathematically, if u and v are solution of the PDE, the so is w = αu + βv where α and β are constants independent
of u, x and y . The Laplace equation
uxx + uyy = 0
is linear while the one dimensional Burger equation
ut + uux = 0
is nonlinear. The majority of the numerical analysis results are valid for linear
equations, and little is known or generalizable to nonlinear equations.
17 18 CHAPTER 2. BASICS OF PDES Quasi Linear A PDE is quasi linear if it is linear in its highest order derivative, i.e.
the coeﬃcients of the PDE multiplying the highest order derivative depends at
most on the function and lower order derivative. Thus, in our example, a, b and c
may depend on u, ux and uy but not on uxx , uyy or uxy . Quasi linear equations
form an important subset of the larger nonlinear class because methods of analysis
and solutions developed for linear equations can safely be applied to quasi linear
equations. The vorticity transport equation of quasigeostrophic motion:
∂ ∇2 ψ
+
∂t ∂ ψ ∂ ∇2 ψ ∂ψ ∂ ∇2 ψ
−
∂y ∂x
∂x ∂y = 0, (2.2) where ∇2 = ψxx + ψyy is a third order quasi linear PDE for the streamfunction ψ . 2.1 Classiﬁcation of second order PDEs The following classiﬁcation are rooted in the character of the physical laws represented
by the PDEs. However, these characteristics can be given a deﬁnite mathematical classﬁcation that at ﬁrst sight has nothing to do with their origins. We will attempt to link
the PDE’s category to the relevant physical roots.
The ﬁrst question in attempting to solve equation 2.1 is to attempt a transformation
of coordinates (the independent variables x and y ) in order to simpplify the equation.
The change of variable can take the general form:
x = x(ξ, η )
y = y (ξ, η ) (2.3) where ξ and η are the new independent variables. This is equivalent to a change of
coordinates. Using the chain rule of diﬀerentiation we have:
ux = uξ ξx + uη ηx (2.4) uy = uξ ξy + uη ηy (2.5) uxx =
uyy = 2
uξξ ξx
2
uξξ ξy +
+ 2
2uξη ξx ηx + uηη ηx + uξ ξxx + uη ηxx
2
2uξη ξy ηy + uηη ηy + uξ ξyy + uη ηyy uxy = uξξ ξx ξy + uξη (ξx ηy + ξy ηx ) + uηη ηx ηy + uξ ξxy + uη ηxy (2.6)
(2.7)
(2.8) Substituting these expression in PDE 2.1 we arrive at the following equation:
Auξξ + Buξη + Cuηη + Duξ + Euη + F = 0, (2.9) where the coeﬃcients are given by the following expressions:
2
2
A = aξx + bξx ξy + cξy (2.10) B = 2aξx ηx + b(ξx ηy + ξy ηx ) + 2cξy ηy (2.11) C= 2
aηx + bηx ηy + 2
cηy (2.12) D = dξx + eξy (2.13) E = dηx + eηy (2.14) F (2.15) =f 2.1. CLASSIFICATION OF SECOND ORDER PDES 19 The equation can be simpliﬁed if ξ and η can be chosen such that A = C = 0 which
in terms of the transformation factors requires:
2
2
aξx + bξx ξy + cξy = 0
2 + bη η + cη 2 = 0
aηx
xy
y (2.16) Assuming ξy and ηy are not equal to zero we can rearrange the above equation to have
the form ar 2 + br + c = 0 where r = ξx /ξy or ηx /ηy . The number of roots for this
quadratic depends on the sign of the determinant b2 − 4ac. Before considering the
diﬀerent cases we note that the sign of the determinant is independent of the choice of
the coordinate system. Indeed it can be easily shown that the determinant in the new
system is B 2 −4AC = (b2 −4ac)(ξx ηy −ξy ηx )2 , and hence the same sign as the determinant
in the old system since the quantity (ξx ηy − ξy ηx ) is nothing but the squared Jacobian
of the mapping between (x, y ) and (ξ, η ) space, and the Jacobian has to be onesigned
for the mapping to be valid. 2.1.1 Hyperbolic Equation: b2 − 4ac > 0 In the case where b2 − 4ac > 0 equation has two√
distincts real roots and the equation is
2
called hyperbolic. The roots are given by r = −b± 2b −4ac . The coordinate transformation
a
required is hence given by:
√
−b + b2 − 4ac
ξx
=
(2.17)
ξy
2a
√
ηx
−b − b2 − 4ac
=
(2.18)
ηy
2a
The interpretation of the above relation can be easily done by focussing on constant ξ
surfaces where dξ = ξx dx + ξy dy = 0, and hence:
dy
dx
dy
dx ξx
=
ξy
ξ
ηx
=− =
ηy
η
=− √
b− b2 −4ac
2a (2.19) √
b+ b2 −4ac
2a (2.20) The roots of the quadratic are hence nothing but the slope of the constant ξ and constant
η curves. These curves are called the characteristic curves. They are the preferred
direction along which information propagate in a hyperbolic system.
In the (ξ, η ) system the equation takes the canonical form:
Buξη + Duξ + Euη + F = 0 (2.21) The solution can be easily obtained in the case D = E = F = 0, and takes the form:
u = G(ξ ) + H (η )
where G and H are function determined by the boundary and initial conditions. (2.22) 20 CHAPTER 2. BASICS OF PDES Example 1 The onedimensional wave equation:
utt − κ2 uxx = 0, −∞ ≤ x ≤ ∞ (2.23) where κ is the wave speed is an example of a hyperbolic system, since its b2 − 4ac =
4κ2 > 0. The slope of the charasteristic curves are given by
dx
= ±κ,
dt (2.24) which, for constant κ, gives the two family of characteristics:
ξ = x − κt, η = x + κt (2.25) Initial conditions are needed to solve this problem; assuming they are of the form:
u(x, 0) = f (x), ut (x, 0) = g(x), (2.26) we get the equations:
F (x) + G(x) = f (x)
′ (2.27) ′ −κF (x) + κG (x) = g(x) (2.28) The second equation can be integrated in x to give
x − κ [F (x) − F (x0 )] + κ [G(x) − G(x0 )] = g(α) dα (2.29) x0 where x0 is arbitrary and α an integration variable. We now have two equations in two
unknowns, F and G, and the system can be solved to get:
F (x) =
G(x) = f (x)
1
−
2
2κ
1
f (x)
+
2
2κ x F (x0 ) − G(x0 )
2
x0
x
F (x0 ) − G(x0 )
.
g(α) dα +
2
x0
g(α) dα − (2.30)
(2.31) To obtain the ﬁnal solution we replace x by x − κt in the expression for F and by x + κt
in the expression for G; adding up the resulting expressions the ﬁnal solution to the PDE
takes the form:
u(x, t) = 1
f (x − κt) + f (x + κt)
+
2
2κ x+κt g(τ ) dτ (2.32) x−κt Figure 2.1 shows the solution of the wave equation for the case where κ = 1, g = 0, and
f (x) is a square wave. The time increases from left to right. The succession of plots
shows two travelling waves, going in the positive and negative xdirection respectively
at the speed κ = 1. Notice that after the crossing, the two square waves travel with no
change of shape. 2.1. CLASSIFICATION OF SECOND ORDER PDES 21 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 0 5 −5 0 5 −5 1 1
0.8
0.6
0.4 0.4 0.2 0.2 0.2 0 0 5 0.6 0.4 0 0.8 0.6 5 1 0.8 0 0 −5 0 5 −5 0 5 −5 Figure 2.1: Solution to the second order wave equation. The top left ﬁgure shows the
initial conditions, and the subsequent ones the solution at t = 0.4, 0.8, 1.2, 1.6 and 2.0. 2.1.2 Parabolic Equation: b2 − 4ac = 0 If b2 − 4ac = 0 then there is only one double root, and the equation is called parabolic.
The two characteristic curves then coincide:
dy
dx =
ξ dy
dx =
η −b
2a (2.33) Since the two characteristic curves coincide, the Jacobian of the mapping ξx ηy − ξy ηx
vanishes identically. The coeﬃcients of the PDE become A = B = 0. The canonical
form of the parabolic equation is then
Cuηη + Duξ + Euη + F = 0 (2.34) Example 2 The heat equation is a common example of parabolic equations:
ut = κ2 uxx (2.35) 22 CHAPTER 2. BASICS OF PDES where κ now stands for a diﬀusion coeﬃcient. The solution in an inﬁnite domain can be
obtained by Fourier transforming the PDE to get:
ut = −k2 κ2 u
˜
˜ (2.36) where u is the Fourier transform of u:
˜
∞ u(k, t) =
˜ −∞ u(x, t)e−ikx dx, (2.37) and k is the wavenumber. The transformed PDE is simply a ﬁrst order ODE, and can
be easily solved with the help of the initial conditions u(x, 0) = u0 (x):
u(k, t) = u0 e−k
˜
˜ 2 κ2 t (2.38) The ﬁnal solution is obtained by back Fourier transform u(x, t) = F−1 (˜). The latu
ter can be written as a convolution since the back transforms of u0 = F−1 (˜0 ), and
u
F−1 (e−k 2 κ2 t )= 2κ 1
√ −x 2 e 4κ2 t are known:
πt
u(x, t) = 1
√
2κ πt ∞
−∞ u0 (X )e −(x −X )2
4κ2 t dX (2.39) 1 0.1 0.01
0.2
0.8 0.6 0.4 0.2 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 Figure 2.2: Solution to the second order wave equation. The ﬁgure shows the initial
conditions (the top hat proﬁle), and the solution at times t = 0.01, 0.1, 0.2.
As an example we show the solution of the heat equation using the same square initial
condition as for the wave equation. The solution can be written in terms of the error
function:
1 −(x −X )2
x−1
1
1
x+1
√ − erf
√
,
(2.40)
e 4κ2 t dX =
u(x, t) = √
erf
2
2κ πt −1
2κ t
2κ t
2 z
1
where erf(z ) = √π 0 e−s ds. The solution is shown in ﬁgure 2.2. Instead of a travelling
wave the solution shows a smearing of the two discontinuities as time increases accompanied by a decrease of the maximum amplitude. As its name indicates, the heat equation
is an example of a diﬀusion equation where gradients in the solution tend to be smeared. 2.2. WELLPOSED PROBLEMS 2.1.3 23 Elliptic Equation: b2 − 4ac < 0 If b2 − 4ac < 0 and there is no real roots; the equation is then called elliptic. There are
no real transformation that can eliminate the second derivatives in ξ and η . However, it
is possible to ﬁnd a transformation that would set B = 0 and A = C = 1. The canonical
form is then:
uξξ + uηη + Duξ + Euη + F = 0
(2.41) Example 3 The Poisson equation in two space dimensions is an example of elliptic
PDE:
uxx + uyy = 0
(2.42) 2.2 WellPosed Problems Before attempting to compute numerical a solution to a PDE, we need to ﬁnd out if its
analytical solution makes sense. In other words, we want to ﬁnd out if enough information
is present in the problem formulation to identify the solution. The deﬁnition of a wellposed problem addresses this issue. A wellposed problem is deﬁned as where the solution
satisﬁes the following properties:
• the solution exists
• the solution is unique
• the solution depends continuously upon the data
Example 4 Consider the system of equations:
uy + vx
=0
vy + γux = 0 (2.43) The system can be reduced to a second order PDE in 1 variable:
uyy − γuxx = 0
vyy − γvxx = 0 (2.44) Clearly, the PDEs are hyperbolic if γ > 0 and elliptic if γ < 0. To solve this system
of equation we require boundary conditions. To continue with our example, we look for
periodic solutions in x and impose the following boundary condition:
u(x, 0) = sin(N x)
, v (x, 0) = 0
N2 (2.45) 1. Ellipitic γ = −β 2 < 0
The solution for this case is then:
u(x, y ) = sin(N x)
cosh(βN y ),
N2 v (x, y ) = β cos(N x)
sinh(βN y )
N2 (2.46) 24 CHAPTER 2. BASICS OF PDES
Notice that even though the PDEs are identical, the two solutions u and v are
diﬀerent because of the boundary conditions. For N → ∞ the boundary conditions
become identical, and hence one would expect the solution to become identical.
However, it is easy to verify that u − v  → ∞ for any ﬁnite y > 0 as N → ∞.
Hence small changes in the boundary condition lead to large changes in the solution,
and the continuity between the boundary data and the solution has been lost. The
problem in this case is that no boundary condition has been imposed on y → ∞
as required by the elliptic nature of the problem. 2. Hyperbolic γ = β 2 > 0
The solution is then
u(x, y ) = sin(N x)
cos(βN y ),
N2 v (x, y ) = −γ cos(N x)
sin(βN y )
N2 (2.47) Notice that in this case u, v → 0 when N → ∞ for any ﬁnite value of y . 2.3 First Order Systems The previous classiﬁcation considered a single, scalar (1 dependent variable), second order
partial diﬀerential equation, and two independent variables. In most physical systems
there are usually a number of PDEs that must be satisﬁed simultaneously involving higher
order derivative. We must then be able to classify these systems before attempting their
solutions. Since systems of PDEs can be recast into ﬁrst order system, the classiﬁcation
uses the latter approach. For example, the wave equation of the previous section can be
cast as two equations in two unknowns:
vt = κηx
ηt = κvx ←→ ∂
∂t v
η = 0κ
κ0 ∂
∂x v
η (2.48) where we have deﬁned η = κux , and v = ut . Note that it is possible to consider v and η
as the component of a vector unknown w and to write the equations in vector notation as
shown in equation (2.48). In the next few section we will look primarily on the condition
under which a ﬁrst order system is hyperbolic. 2.3.1 Scalar Equation A typical equation of this sort is the advection equation:
ut + cux = 0, 0 ≤ x ≤ L
u(x, t = 0) = u0 (x), u(x = 0, t) = ul (t) (2.49) where c is the advecting velocity. Let us deﬁne c = dx/dt, so that the equation becomes:
∂u
∂u
dt +
dx = du = 0
∂t
∂x (2.50) 2.3. FIRST ORDER SYSTEMS 25 1.4
u(x,t)=u (x )
0 1.2 0 1 t 0.8
0.6
0.4
u (x )
00 0.2
0
−4 −2 0 2 4 6 x Figure 2.3: Characteristic lines for the linear advection equation. The solid lines are the
characteristics emanating from diﬀerent locations on the initial axis. The dashed line
represents the signal at time t = 0 and t = 1. If the solution at (x, t) is desired, we ﬁrst
need to ﬁnd the foot of the characteristic x0 = x − ct, and the value of the function there
at the initial time is copied to time t.
where du is the total diﬀerential of the function u. Since the right hand side is zero,
then the function must be constant along the lines dx/dt = c, and this constant must be
equal to the value of u at the initial time. The solution can then written as:
u(x, t) = u0 (x0 ) along dx
=c
dt (2.51) where x0 is the location of the foot of the characteristic, the intersection of the characteristic with the t = 0 line. The simplest way to visualize this picture is to consider
the case where c is constant. The characteristic lines can then be obtained analytically:
they are straight lines given by x = x0 + ct. A family of characteristic lines are shown
in ﬁgure 2.3.1 where c is assumed positive. In this example the information is travelling
from left to right at the constant speed c, and the initial hump translates to the right
without change of shape.
If the domain is of ﬁnite extent, say 0 ≤ x ≤ L, and the characteristic intersects the
line x = 0 (assuming c > 0), then a boundary condition is needed on the left boundary
to provide the information needed to determine the solution uniquely. That is we need
to provide the variation of u at the “inlet” boundary in the form:u(x = 0, t) = g(t). The
solution now can be written as:
u(x, t) = u0 (x − ct)
g(t − x/c) for
for x − ct > 0
x − ct < 0 (2.52) Not that since the information travels from left to right, the boundary condition is
needed at the left boundary and not on the right. The solution would be impossible
to determine had the boundary conditions been given on the right boundary x = L, 26 CHAPTER 2. BASICS OF PDES 2
1.5 2
u(x,0)=1−sin(π x) 1.5 1 1 0.5 0.5 0
−1 −0.5 0 0.5 0
−1 1 −0.5 0 0.5 1 Figure 2.4: Characteristics for Burgers’ equation (left panel), and the solution (right
panel) at diﬀerent times for a periodic problem. The black line is the initial condition,
the red line the solution at t = 1/8, the blue at t = 1/4, and the magenta at t = 3/4.
The solution become discontinuous when characteristics intersects.
the problem would then be illposed for lack of proper boundary conditions. The right
boundary maybe called an “outlet” boundary since information is leaving the domain.
No boundary condition can be imposed there since the solution is determined from
“upstream” information. In the case where c < 0 the information travels from right to
left, and the boundary condition must be imposed at x = L.
If the advection speed c varies in x and t, then the characteristics are not necessarily
straight lines of constant slope, but are in general curves. Since the slopes of the curves
vary, characteristic lines may intersects. These intersection points are places where the
solution is discontinuous with sharp jumps. At these locations the slopes are inﬁnite and
space and timederivative become meaningless, i.e. the PDE is not valid anymore. This
breakdown occurs usually because of the neglect of important physical terms, such as
dissipative terms, that act to prevent true discontinuous solutions.
An example of an equation that can lead to discontinuous solutions is the Burger
equation:
ut + uux = 0
(2.53)
where c = u. This equation is nonlinear with a variable advection speed that depend on
the solution. The characteristics are given by the lines:
dx
=u
dt (2.54) along which the PDE takes the form du = 0. Hence u is constant along characteristics,
which means that their slopes are also constant according to equation (2.54), and hence
must be straightlines. Even in this nonlinear case the characteristics are straightlines
but with varying slopes. The behavior of the solution can become quite complicated as
characteristic lines intersect as shown in ﬁgure 2.4. The solution of hyperbolic equations
in the presence of discontinuities can become quite complicated. We refer the interested
reader to Whitham (1974); Durran (1999) for further discussions. 2.3. FIRST ORDER SYSTEMS 2.3.2 27 System of Equations in onespace dimension A system of PDE in onespace dimension of the form
∂w
∂w
+A
=0
∂t
∂x (2.55) where A is the so called Jacobian matrix is said to be hyperbolic if the matrix A has
a complete set of real eigenvalues. For then one can ﬁnd a bounded matrix T whose
columns are the eigenvectors of A, such that the matrix D = T−1 AT is diagonal.
Reminder: A diagonal matrix is one where all entries are zero save for the ones on the
main diagonal, and in the case above the diagonal entries are the eigenvalues of the
ˆ
matrix. The system can then be uncoupled by deﬁning the auxiliary variables w = Tw,
replacing in the original equation we get the equations
ˆ
ˆ
∂w
∂ wi
ˆ
∂ wi
ˆ
∂w
+D
= 0 ←→
+ λi
=0
∂t
∂x
∂t
∂x (2.56) The equation on the left side is written in the vector form whereas the component form
on the right shows the uncoupling of the equations. The component form clearly shows
how the sytem is hyperbolic and analogous to the scalar form.
Example 5 The linearized equation governing tidal ﬂow in a channel of constant cross
section and depth are the shallow water equations:
∂
∂t u
η + 0g
h0 ∂
∂x u
η = 0, A= 0g
h0 (2.57) where u and η are the unknown velocity and surface elevation, g is the gravitational
acceleration, and h the water depth. The eigenvalues of the matrix A can be found by
solving the equation:
det A = 0 ⇔ det −λ g
h −λ = λ2 − gh = 0. (2.58) √
The two real roots of the equations are λ = ±c, where c = gh. Hence the eigenvalues
are the familiar gravity wave speed. Two eigenvectors of the system are u1 and u2
corresponding to the positive and negative roots, respectively: 1
1
u1 = c , u2 = − c .
g
g (2.59) The eigenvectors are the columns of the transformation matrix T, and we have
T= 1
c
g 1
c
−g , T −1 = 1
2 g
c
−g
c 1
1 . (2.60) It it easy to verify that
D = T−1 AT = c0
0 −c , (2.61) 28 CHAPTER 2. BASICS OF PDES
t T u(x, t) d d d d d d
1
−1 d c
c d d d
u(x, t) = u0 (xa )
ˆ
ˆ
η (x, t) = η0 (xb ) d
ˆ
ˆ d xa E xb x Figure 2.5: Characteristics for the onedimensional tidal equations. The new variables
u and η are constant along the right, and left going characteristic, respectively. The
ˆ
ˆ
solution at the point (x, t) can be computed by ﬁnding the foot of two characteristic
curves at the initial time, xa and xb and applying the conservation of u and η .
ˆ
ˆ
as expected. The new variable are given by
u
ˆ
η
ˆ = T −1 u
η = 1
2 1
1 g
c
−g
c u
η gη
1 2 u+ c
= 1 gη
u−
2
c (2.62) The equations written in the new variables take the component form: ˆ
ˆ ∂u + c ∂u ∂t ∂x ˆ ∂η ˆ − c ∂η
∂t ∂x =0
=0 ←→ u = constant along
ˆ η = constant along
ˆ dx
=c
dt
dx
= −c
dt (2.63) To illustrate how the characteristic information can be used to determine the solution
at any time, given the initial conditions, we consider the case of an inﬁnite domain, and
calculate the intersection of the two characteristic curves meeting at (x, t) with the axis
t = 0. If we label the two coordinate xa and xb as shown in the ﬁgure, then we can set
up the two equations: gη u+ c u − gη
c 2.3.3 = ua +
= ub − gηa
c
gη
b c u = ua + ub
ηa − ηb
+g
2
2c
←→ η = c ua − ub + ηa + ηb 2g
2 (2.64) System of equations in multispace dimensions A system of PDE in twospace dimension can be written in the form:
∂w
∂w
∂w
+A
+B
=0
∂t
∂x
∂y (2.65) 2.3. FIRST ORDER SYSTEMS 29 In general it is not possible to deﬁne a similarity transformation that would diagonalize
all matrices at once. To extend the deﬁnition of hyperbolicity to multispace dimension,
the following procedure can be used for the case where A and B are constant matrices.
First we deﬁne the Fourier transform of the dependent variables with respect to x, y and
t:
ˆ
w = wei(kx+ly−ωt)
(2.66)
ˆ
where w can be interpreted as the vector of Fourier amplitudes, and k and l are wavelength in the x and y direction respectively, and ω is the frequency. This Fourier representation is then substitute it in the partial diﬀerential equation to obtain:
ˆ
[kA + lB − ωI ] w = 0, (2.67) where I is the identity matrix. Equation (2.67) has the form of an eigenvalue problem,
where ω represent the eigenvalues of the matrix kA + lB. The system is classiﬁed as
hyperbolic if and only if the eigenvalues ω are real for real choices of the wavenumber
vector (k, l). The extension above hinges on the matrices A and B being constant in
space and time; in spite of its limitation it does show intuitively how the general behavior
of wavelike solution can be extended to multiple spatial dimensions.
For the case where the matrices are not constant, the deﬁnition can be extended by
requiring that the exitence of bounded matrices T such that the matrix T−1 (kA + lB)T
is a diagonal matrix with real eigenvalues for all points within an neighborhood of (x, y ). 30 CHAPTER 2. BASICS OF PDES Chapter 3 Finite Diﬀerence Approximation
of Derivatives
3.1 Introduction The standard deﬁnition of derivative in elementary calculus is the following
u(x + ∆x) − u(x)
∆x→ 0
∆x u′ (x) = lim (3.1) Computers however cannot deal with the limit of ∆x → 0, and hence a discrete analogue
of the continuous case need to be adopted. In a discretization step, the set of points on
which the function is deﬁned is ﬁnite, and the function value is available on a discrete set
of points. Approximations to the derivative will have to come from this discrete table of
the function.
Figure 3.1 shows the discrete set of points xi where the function is known. We
will use the notation ui = u(xi ) to denote the value of the function at the ith node
of the computational grid. The nodes divide the axis into a set of intervals of width
∆xi = xi+1 − xi . When the grid spacing is ﬁxed, i.e. all intervals are of equal size, we
will refer to the grid spacing as ∆x. There are deﬁnite advantages to a constant grid
spacing as we will see later. 3.2 Finite Diﬀerence Approximation The deﬁnition of the derivative in the continuum can be used to approximate the derivative in the discrete case:
u′ (xi ) ≈ ui+1 − ui
u(xi + ∆x) − u(xi )
=
∆x
∆x (3.2) where now ∆x is ﬁnite and small but not necessarily inﬁnitesimally small, i.e. . This is
known as a forward Euler approximation since it uses forward diﬀerencing. Intuitively,
the approximation will improve, i.e. the error will be smaller, as ∆x is made smaller.
31 32 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES x i−1 x i x i+1 Figure 3.1: Computational grid and example of backward, forward, and central approximation to the derivative at point xi . The dashdot line shows the centered parabolic
interpolation, while the dashed line show the backward (blue), forward (red) and centered
(magenta) linear interpolation to the function.
The above is not the only approximation possible, two equally valid approximations are:
backward Euler:
u(xi ) − u(xi − ∆x)
ui − ui−1
u′ (xi ) ≈
=
(3.3)
∆x
∆x
Centered Diﬀerence
u′ (xi ) ≈ u(xi + ∆x) − u(xi − ∆x)
ui+1 − ui−1
=
2∆x
2∆x (3.4) All these deﬁnitions are equivalent in the continuum but lead to diﬀerent approximations
in the discrete case. The question becomes which one is better, and is there a way to
quantify the error committed. The answer lies in the application of Taylor series analysis.
We brieﬂy describe Taylor series in the next section, before applying them to investigate
the approximation errors of ﬁnite diﬀerence formulae. 3.3. TAYLOR SERIES 3.3 33 Taylor series Starting with the identity:
x u(x) = u(xi ) + xi u′ (s) ds (3.5) Since u(x) is arbitrary, the formula should hold with u(x) replaced by u′ (x), i.e.,
u′ (x) = u′ (xi ) + x
xi u′′ (s) ds (3.6) Replacing this expression in the original formula and carrying out the integration (since
u(xi ) is constant) we get:
u(x) = u(xi ) + (x − xi )u′ (xi ) + x x xi xi u′′ (s) ds ds (3.7) The process can be repeated with
u′′ (x) = u′′ (xi ) + x
xi u′′′ (s) ds (3.8) to get:
u(x) = u(xi ) + (x − xi )u′ (xi ) + (x − xi )2 ′′
u (xi ) +
2! x x x xi xi xi u′′′ (s) ds ds ds (3.9) This process can be repeated under the assumption that u(x) is suﬃciently diﬀerentiable, and we ﬁnd:
(x − xi )n (
(x − xi )2 ′′
u (xi ) + · · · +
u n)(xi ) + Rn+1 (3.10)
2!
n!
where the remainder is given by:
u(x) = u(xi ) + (x − xi )u′ (xi ) + x x Rn+1 = xi ··· u(n+1) (x)( ds)n+1 (3.11) xi Equation 3.10 is known as the Taylor series of the function u(x) about the point xi .
Notice that the series is a polynomial in (x − xi ) (the signed distance of x to xi ), and
the coeﬃcients are the (scaled) derivatives of the function evaluated at xi .
If the (n + 1)th derivative of the function u has minimum m and maximum M over
the interval [xi x] then we can write:
x x
xi ··· xi m m( ds)n+1 ≤ Rn+1 ≤ (x − xi )n+1
(n + 1)! x x
xi ≤ Rn+1 ≤ M ··· M ( ds)n+1 (3.12) xi (x − xi )n+1
(n + 1)! (3.13) which shows that the remainder is bounded by the values of the derivative and the
distance of the point x to the expansion point xi raised to the power (n + 1). If we
further assume that u(n+1) is continuous then it must take all values between m and M
that is
(x − xi )n+1
(3.14)
Rn+1 = u(n+1) (ξ )
(n + 1)!
for some ξ in the interval [xi x]. 34 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES 3.3.1 Taylor series and ﬁnite diﬀerences Taylor series have been widely used to study the behavior of numerical approximation
to diﬀerential equations. Let us investigate the forward Euler with Taylor series. To do
so, we expand the function u at xi+1 about the point xi :
u(xi + ∆xi ) = u(xi ) + ∆xi ∂u
∂x +
xi ∆ x2 ∂ 2 u
i
2! ∂x2 +
xi ∆ x3 ∂ 3 u
i
3! ∂x3 + ... (3.15) xi The Taylor series can be rearranged to read as follows:
∂u
u(xi + ∆xi ) − u(xi )
−
∆ xi
∂x =
xi ∆ xi ∂ 2 u
2! ∂x2 ∆ x2 ∂ 3 u
i
3! ∂x3 +
xi + ... (3.16) xi Truncation Error
where it is now clear that the forward Euler formula (3.2) corresponds to truncating the
Taylor series after the second term. The right hand side of equation (3.16) is the error
committed in terminating the series and is referred to as the truncation error. The
tuncation error can be deﬁned as the diﬀerence between the partial derivative and its
ﬁnite diﬀerence representation. For suﬃciently smooth functions, i.e. ones that possess
continuous higher order derivatives, and suﬃciently small ∆xi , the ﬁrst term in the series
can be used to characterize the order of magnitude of the error. The ﬁrst term in the
truncation error is the product of the second derivative evaluated at xi and the grid
spacing ∆xi : the former is a property of the function itself while the latter is a numerical
2
parameter which can be changed. Thus, for ﬁnite ∂ u , the numerical approximation
∂x2
depends lineraly on the parameter ∆xi . If we were to half ∆xi we ought to expect a
linear decrease in the error for suﬃciently small ∆xi . We will use the “big Oh” notation
to refer to this behavior so that T.E. ∼ O(∆xi ). In general if ∆xi is not constant we
pick a representative value of the grid spacing, either the average of the largest grid
spacing. Note that in general the exact truncation error is not known, and all we can do
is characterize the behavior of the error as ∆x → 0. So now we can write:
∂u
∂x =
xi ui+1 − ui
+ O(∆x)
∆ xi (3.17) The taylor series expansion can be used to get an expression for the truncation error
of the backward diﬀerence formula:
u(xi − ∆xi−1 ) = u(xi ) − ∆xi−1 ∂u
∂x +
xi ∆x2−1 ∂ 2 u
i
2! ∂x2 xi − ∆x3−1 ∂ 3 u
i
3! ∂x3 + ... (3.18) xi where ∆xi−1 = xi − xi−1 . We can now get an expression for the error corresponding to
backward diﬀerence approximation of the ﬁrst derivative:
∂u
u(xi ) − u(xi − ∆xi−1 )
−
∆xi−1
∂x xi =− ∆xi−1 ∂ 2 u
2! ∂x2 +
xi ∆x2−1 ∂ 3 u
i
3! ∂x3 Truncation Error + ...
xi (3.19) 3.3. TAYLOR SERIES 35 It is now clear that the truncation error of the backward diﬀerence, while not the same
as the forward diﬀerence, behave similarly in terms of order of magnitude analysis, and
is linear in ∆x:
∂u
ui − ui−1
+ O(∆x)
(3.20)
=
∂x xi
∆ xi
Notice that in both cases we have used the information provided at just two points to
derive the approximation, and the error behaves linearly in both instances.
Higher order approximation of the ﬁrst derivative can be obtained by combining the
two Taylor series equation (3.15) and (3.18). Notice ﬁrst that the high order derivatives
of the function u are all evaluated at the same point xi , and are the same in both
expansions. We can now form a linear combination of the equations whereby the leading
error term is made to vanish. In the present case this can be done by inspection of
equations (3.16) and (3.19). Multiplying the ﬁrst by ∆xi−1 and the second by ∆xi and
adding both equations we get:
ui − ui−1
ui+1 − ui
∂u
1
+ ∆xi
∆xi−1
−
∆xi + ∆xi−1
∆ xi
∆xi−1
∂x =
xi ∆xi−1 ∆xi ∂ 3 u
3!
∂x3 + ...
xi (3.21)
There are several points to note about the preceding expression. First the approximation
uses information about the functions u at three points: xi−1 , xi and xi+1 . Second the
truncation error is T.E. ∼ O(∆xi−1 ∆xi ) and is second order, that is if the grid spacing is
decreased by 1/2, the T.E. error decreases by factor of 22 . Thirdly, the previous point can
be made clearer by focussing on the important case where the grid spacing is constant:
∆xi−1 = ∆xi = ∆x, the expression simpliﬁes to:
ui+1 − ui−1
∂u
−
2∆x
∂x =
xi ∆x2 ∂ 3 u
3! ∂x3 + ... (3.22) xi Hence, for an equally spaced grid the centered diﬀerence approximation converges quadratically as ∆x → 0:
ui+1 − ui−1
∂u
+ O(∆x2 )
(3.23)
=
∂x xi
2∆x
Note that like the forward and backward Euler diﬀerence formula, the centered diﬀerence uses information at only two points but delivers twice the order of the other two
methods. This property will hold in general whenever the grid spacing is constant and
the computational stencil, i.e. the set of points used in approximating the derivative, is
symmetric. 3.3.2 Higher order approximation The Taylor expansion provides a very useful tool for the derivation of higher order approximation to derivatives of any order. There are several approaches to achieve this.
We will ﬁrst look at an expendient one before elaborating on the more systematic one.
In most of the following we will assume the grid spacing to be constant as is usually the
case in most applications. 36 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES Equation (3.22) provides us with the simplest way to derive a fourth order approximation. An important property of this centered formula is that its truncation error
contains only odd derivative terms:
∂u ∆x2 ∂ 3 u ∆x4 ∂ 5 u ∆x6 ∂ 7 u
∆x2m ∂ (2m+1) u
ui+1 − ui−1
=
+
+
+
+ ... +
+ ...
2∆x
∂x
3! ∂x3
5! ∂x5
7! ∂x7
(2m + 1)! ∂x(2m+1)
(3.24)
The above formula can be applied with ∆x replace by 2∆x, and 3∆x respectively to get:
ui+2 − ui−2
4∆x
ui+3 − ui−3
6∆x =
= ∂u (2∆x)2 ∂ 3 u (2∆x)4 ∂ 5 u (2∆x)6 ∂ 7 u
+
+
+
+ O(∆x8 ) (3.25)
∂x
3! ∂x3
5! ∂x5
7! ∂x7
∂u (3∆x)2 ∂ 3 u (3∆x)4 ∂ 5 u (3∆x)6 ∂ 7 u
+
+
+
+ O(∆x8 ) (3.26)
∂x
3! ∂x3
5! ∂x5
7! ∂x7 It is now clear how to combine the diﬀerent estimates to obtain a fourth order approximation to the ﬁrst derivative. Multiplying equation (3.24) by 22 and substracting it from
equation (3.25), we cancel the second order error term to get:
∂u 4∆x4 ∂ 5 u 20∆x6 ∂ 7 u
8(ui+1 − ui−1 ) − (ui+2 − ui−2 )
=
−
−
+ O(∆x8 )
12∆x
∂x
5! ∂x5
7! ∂x7 (3.27) Repeating the process for equation but using the factor 32 and substracting it from
equation (3.26), we get
∂u 9∆x4 ∂ 5 u 90∆x6 ∂ 7 u
27(ui+1 − ui−1 ) − (ui+3 − ui−3 )
=
−
−
+ O(∆x8 )
48∆x
∂x
5! ∂x5
7! ∂x7 (3.28) Although both equations (3.27) and (3.28) are valid, the latter is not used in practice
since it does not make sense to disregard neighboring points while using more distant
ones. However, the expression is useful to derive a sixth order approximation to the ﬁrst
derivative: multiply equation (3.28) by 9 and equation (3.28) by 4 and substract to get:
45(ui+1 − ui−1 ) − 9(ui+2 − ui−2 ) + (ui+3 − ui−3 )
∂u 36∆x6 ∂ 7 u
=
+
+ O(∆x8 ) (3.29)
60∆x
∂x
7! ∂x7
The process can be repeated to derive higher order approximations. 3.3.3 Remarks The validity of the Taylor series analysis of the truncation error depends on the existence
of higher order derivatives. If these derivatives do not exist, then the higher order
approximations cannot be expected to hold. To demonstrate the issue more clearly we
will look at speciﬁc examples.
Example 6 The function u(x) = sin πx is inﬁnitely smooth and diﬀerentiable, and its
ﬁrst derivative is given by ux = π cos πx. Given the smoothness of the function we expect
the Taylor series analysis of the truncation error to hold. We set about verifying this claim
in a practical calculation. We lay down a computational grid on the interval −1 ≤ x ≤ 1
of constant grid spacing ∆x = 2/M . The approximation points are then xi = i∆x − 1, 3.3. TAYLOR SERIES 37 1 0.01 0.5 0.005 0 0 −0.5 −0.005 −1
−1 −0.5 0 0.5 −0.01
−1 1 0 10 10 −5 10 10 −10 10 10 −15 10 0 10 10 1 2 10 3 10 4 10 10 −0.5 0 0.5 1 0 −5 −10 −15 10 0 1 10 2 10 3 10 4 10 Figure 3.2: Finite diﬀerence approximation to the derivative of the function sin πx. The
top left panel shows the function as a function of x. The top right panel shows the
spatial distribution of the error using the Forward diﬀerence (black line), the backward
diﬀerence (red line), and the centered diﬀerences of various order (magenta lines) for the
case M = 1024; the centered diﬀerence curves lie atop each other because their errors
are much smaller then those of the ﬁrst order schemes. The lower panels are convergence
curves showing the rate of decrease of the rms and maximum errors as the number of
grid cells increases. 38 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES i = 0, 1, . . . , M . Let ǫ be the error between the ﬁnite diﬀerence approximation to the
ﬁrst derivative, ux , and its analytical derivative ux :
˜
ǫi = ux (xi ) − ux (xi )
˜ (3.30) The numerical approximation ux will be computed using the forward diﬀerence, equation
˜
(3.17), the backward diﬀerence, equation (3.20), and the centered diﬀerence approximations of order 2, 4 and 6, equations (3.22), (3.27, and (3.29). We will use two measures
to characterize the error ǫi , and to measure its rate of decrease as the number of grid
points is increased. One is a bulk measure and consists of the root mean square error,
and the other one consists of the maximum error magnitude. We will use the following
notations for the rms and max errors:
1
2 M ǫ 2 ǫ2
i (3.31) max (ǫi ) (3.32) = ∆x
i=0 ǫ ∞ = 0≤i≤M The right panel of ﬁgure 3.2 shows the variations of ǫ as a function of x for the case
M = 1024 for several ﬁnite diﬀerence approximations to ux . For the ﬁrst order schemes
the errors peak at ±1/2 and reaches 0.01. The error is much smaller for the higher order
centered diﬀerence scheme. The lower panels of ﬁgure 3.2 show the decrease of the rms
error ( ǫ 2 on the left), and maximum error ( ǫ ∞ on the right) as a function of the
number of cells M . It is seen that the convergence rate increases with an increase in
the order of the approximation as predicted by the Taylor series analysis. The slopes
on this loglog plot are 1 for forward and backward diﬀerence, and 2, 4 and 6 for the
centered diﬀerence schemes of order 2, 4 and 6, respectively. Notice that the maximum
error decreases at the same rate as the rms error even though it reports a higher error.
Finally, if one were to gauge the eﬃciency of using information most accurately, it is
evident that for a given M , the high order methods achieve the lowest error.
Example 7 We now investigate the numerical approximation to a function with ﬁnite
diﬀerentiability, more precisely, one that has a discontinuous third derivative. This function is deﬁned as follows:
u(x)
ux (x)
x < 0 sin πx
π cos πx
−x2 π (1 − 2x2 )e−x2
0 < x πxe
x=0
0
π uxx (x)
−π 2 sin πx
2
2πx(2x2 − 3)e−x
0 uxxx
−π 3 cos πx
2
−2π (3 − 12x2 + 4x4 )e−x
−π 3 , −6π Notice that the function and its ﬁrst two derivatives are continuous at x = 0, but the
third derivative is discontinuous. An examination of the graph of the function in ﬁgure
3.3 shows a curve, at least visually (the so called eyeball norm). The error distribution
is shown in the top right panel of ﬁgure 3.3 for the case M = 1024 and the fourth order
centered diﬀerence scheme. Notice that the error is very small except for the spike near
the discontinuity. The error curves (in the lower panels) show that the second order
centered diﬀerence converges faster then the forward and backward Euler scheme, but 3.3. TAYLOR SERIES 39 1.5 1 1 x 10 −6 0.5 0 ε u(x) 0.5
0
−0.5 −0.5
−1
−1 10 10 0
x 0.5 −1
−1 1 0 10 max( ε )  ε  2 10 −0.5 −5 −10 10 0 10 1 2 10
M 10 3 10 4 10 10 −0.5 0
x 0.5 1 0 −5 −10 10 0 10 1 2 10
M 10 3 10 4 Figure 3.3: Finite diﬀerence approximation to the derivative of a function with discontinuous third derivative. The top left panel shows the function u(x) which, to the eyeball
norm, appears to be quite smooth. The top right panel shows the spatial distribution
of the error (M = 1024) using the fourth order centered diﬀerence: notice the spike at
the discontinuity in the derivative. The lower panels are convergence curves showing the
rate of decrease of the rms and maximum errors as the number of grid cells increases. 40 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES that the convergence rates of the fourth and sixth order centered schemes are no better
then that of the second order one. This is a direct consequence of the discontinuity in
the third derivative whereby the Taylor expansion is valid only up to the third term. The
eﬀects of the discontinuity are more clearly seen in the maximum error plot (lower right
panel) then in the mean error one (lower left panel). The main message of this example
is that for functions with a ﬁnite number of derivatives, the Taylor series prediction for
the high order schemes does not hold. Notice that the error for the fourth and sixth
order schemes are lower then the other 3, but their rate of convergence is the same as
the second order scheme. This is largely coincidental and would change according to the
function. 3.3.4 Systematic Derivation of higher order derivative The Taylor series expansion provides a systematic way of deriving approximation to
higher order derivatives of any order (provided of course that the function is smooth
enough). Here we assume that the grid spacing is uniform for simplicity. Suppose that
the stencil chosen includes the points: xj such that i − l ≤ j ≤ i + r . There are thus l
points to the left and r points to the right of the point i where the derivative is desired
for a total of r + l + 1 points. The Taylor expansion is:
(m∆x)2
(m∆x)3
(m∆x)4
(m∆x)5
(m∆x)
ux +
uxx +
uxxx +
uxxxx +
uxxxxx +. . .
1!
2!
3!
4!
5!
(3.33)
for m = −l, . . . , r . Multiplying each of these expansions by a constant am and summing
them up we obtain the following equation: ui+m = ui + r
m=−l,m=0 am ui+m − r m=−l,m=0 am ui = + + + + r
m=−l,m=0
r
m=−l,m=0
r
m=−l,m=0
r
m=−l,m=0
r
m=−l,m=0 + ... mam ∆x ∂u
1! ∂x i ∆x2 ∂ 2 u
2! ∂x2 ∆x3 ∂ 3 u
3! ∂x3 ∆x4 ∂ 4 u
4! ∂x4 ∆x5 ∂ 5 u
5! ∂x5 m2 am m3 am m4 am m5 am i i i i (3.34) It is clear that the coeﬃcient of the kth derivative is given by bk = r =−l,m=0 mk am .
m
Equation (3.34) allows us to determine the r + l coeﬃcients am according to the derivative
desired and the order desired. Hence if the ﬁrst order derivative is needed at fourth order
accuracy, we would set b1 to 1 and b2,3,4 = 0. This would provide us with four equations, 3.3. TAYLOR SERIES 41 and hence we need at least four points in order to determine its solution uniquely. More
generally, if we need the kth derivative then the highest derivative to be neglected must
be of order k + p − 1, and hence k + p − 1 points are needed. The equations will then
have the form:
r bq =
m=−l,m=0 mq am = δqk , q = 1, 2, . . . , k + p − 1 (3.35) where δqk is the Kronecker delta δqk = 1 is q = k and 0 otherwise. For the solution to
exit and be unique we must have: l + r = k + p. Once the solution is obtained we can
determine the leading order truncation term by calculating the coeﬃcient multiplying
the next higher derivative in the truncation error series:
r mk+p am . bk+1 (3.36) m=−l,m=0 Example 8 As an example of the application of the previous procedure, let us ﬁx the
stencil to r = 1 and l = −3. Notice that this is an oﬀcentered scheme. The system of
equation then reads as follows in matrix form: −3
−2
−1
1
a−3
(−3)2 (−2)2 (−1)2 (1)2 a−2 (−3)3 (−2)3 (−1)3 (1)3 a−1
(−3)4 (−2)4 (−1)4 (1)4
a1 = b1
b2
b3
b4 (3.37) If the ﬁrst derivative is desired to fourth order accuracy, we would set b1 = 1 and
b2,3,4 = 0, while if the second derivative is required to third order accuracy we would set
b1,3,4 = 0 and b2 = 1. The coeﬃcients for the ﬁrst example would be: 3.3.5 a−3
a−2
a−1
a1 1 = 12 −1
12
−18
3 (3.38) Discrete Operator Operators are often used to describe the discrete transformations needed in approximating derivatives. This reduces the lengths of formulae and can be used to derive new
approximations. We will limit ourselves to the case of the centered diﬀerence operator:
δnx ui =
δx ui =
δ2x ui = ui+ n − ui− n
2
2 n ∆x
ui+ 1 − ui− 1 2
= ux + O(∆x2 )
∆x
ui+1 − ui−1
= ux + O(∆x2 )
2∆x
2 (3.39)
(3.40)
(3.41) 42 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES
The second order derivative can be computed by noticing that
2
δx ui = δx (δx ui ) = δx (ux + O(∆x2 )
ui+ 1 − ui− 1
2
2
δx
= uxx + O(∆x2 )
∆x 1
δx (ui+ 1 ) − δx (ui− 1 ) = uxx + O(∆x2 )
2
2
∆x
ui+1 − 2ui + ui−1
) = uxx + O(∆x2 )
∆ x2 (3.42)
(3.43)
(3.44)
(3.45) The truncation error can be veriﬁed by going through the formal Taylor series analysis.
Another application of operator notation is the derivation of higher order formula.
For example, we know from the Taylor series that
δ2x ui = ux + ∆ x2
uxxx + O(∆x4 )
3! (3.46) If I can estimate the third order derivative to second order then I can substitute this
2
estimate in the above formula to get a fourth order estimate. Applying the δx operator
to both sides of the above equation we get:
∆ x2
uxxx + O(∆x4 )) = uxxx + O(∆x2 )
3! 2
2
δx (δ2x ui ) = δx (ux + Thus we have
δ2x ui = ux + (3.47) ∆ x2 2
δ [δ2x ui + O(∆x2 )]
3! x (3.48) ∆x3 2
δ δ2x ui + O(∆x4 )
3! x (3.49) Rearranging the equation we have:
ux xi = 3.4 1− Polynomial Fitting Taylor series expansion are not the only means to develop ﬁnite diﬀerence approximation.
An another approach is to rely on polynomial ﬁtting such as splines (which we will
not discuss here), and Lagrange interpolation. We will concentrate on the later in the
following section.
Lagrange interpolation consists of ﬁtting a polynomial of a speciﬁed defree to a given
set of (xi , ui ) pairs. The slope at the point xi is approximated by taking the derivative
of the polynomial at the point. The approach is best illustrate by looking at speciﬁc
examples. 3.4.1 Linear Fit The linear polynomial:
L1 (x) = x − xi+1
x − xi
ui+1 −
ui , xi ≤ x ≤ xi+1
∆x
∆x (3.50) 3.4. POLYNOMIAL FITTING 43 The derivative of this function yields the forward diﬀerence formula:
ux xi = ∂L1 (x)
∂x =
xi ui+1 − ui
∆x (3.51) A Taylor series analysis will show this approximation to be linear. Likewise if a linear
interpolation is used to interpolate the function in xi−1 ≤ x ≤ xi we get the backward
diﬀerence formula. 3.4.2 Quadratic Fit It is easily veriﬁed that the following quadratic interpolation will ﬁt the function values
at the points xi and xi±1 :
(x − xi )(x − xi+1 )
(x − xi−1 )(x − xi+1 )
(x − xi−1 )(x − xi )
ui−1 −
ui +
ui+1
2∆x2
∆ x2
2∆x2
(3.52)
Diﬀerentiating the functions and evaluating it at xi we can get expressions for the ﬁrst
and second derivatives:
L2 (x) = ∂L2
∂x
∂ 2 L2
∂x2 = ui+1 − ui−1
2∆x (3.53) = ui+1 − 2ui + ui−1
∆x2 (3.54) xi xi Notice that these expression are identical to the formulae obtained earlier. A Taylor
series analysis would conﬁrm that both expression are second order accurate. 3.4.3 Higher order formula Higher order fomula can be develop by Lagrange polynomials of increasing degree. A
word of caution is that high order Lagrange interpolation is practical when the evaluation
point is in the middle of the stencil. High order Lagrange interpolation is notoriously
noisy near the end of the stencil when equal grid spacing is used, and leads to the well
known problem of Runge oscillations Boyd (1989). Spectral methods that do not use
periodic Fourier functions (the usual “sin” and “cos” functions) rely on unevenly spaced
points.
To illustrate the Runge phenomenon we’ll take the simple example of interpolating
the function
1
(3.55)
f (x) =
1 + 25x2
in the interval x ≤ 1. The Lagrange interpolation using an equally spaced grid is shown
in the upper panel of ﬁgure 3.4, the solid line refers to the exact function f while the
dashedcolored lines to the Lagrange interpolants of diﬀerent orders. In the center of
the interval (near x = 0, the diﬀerence between the dashed lines and the solid black
line decreases quickly as the polynomial order is increased. However, near the edges
of the interval, the Lagrangian interpolants oscillates between the interpolation points. 44 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES Equally−Spaced Lagrange Interpolation Equally−Spaced Lagrange Interpolation 1 2 0.5 0
0 −2 −0.5 −4
Exact
5 Points
9 Points
13 Points −1.5
−2 f(x) f(x) −1 Exact
17 Points −6
−8 −2.5 −10 −3 −12 −3.5
−4
−1 −14
−0.5 0
x 0.5 1 −16
−1 −0.5 0 0.5 Gauss−Lobatto−Spaced Lagrange Interpolation
1.2
Exact
5 Points
9 Points
13 Points
17 Points 1
0.8 f(x) 0.6
0.4
0.2
0
−0.2
−1 −0.5 0
x 0.5 1 Figure 3.4: Illustration of the Runge phenomenon for equallyspaced Lagrangian interpolation (upper ﬁgures). The right upper ﬁgure illustrate the worsening amplitude of
the oscillations as the degree is increased. The Runge oscillations are suppressed if an
unequally spaced set of interpolation point is used (lower panel); here one based on
GaussLobatto roots of Chebyshev polynomials. The solution black line refers to the
exact solution and the dashed lines to the Lagrangian interpolants. The location of the
interpolation points can be guessed by the crossing of the dashed lines and the solid black
line. 1 3.4. POLYNOMIAL FITTING 45 At a ﬁxed point near the boundary, the oscillations’ amplitude becomes bigger as the
polynomial degree is increased: the amplitude of the 16 order polynomial reaches of value
of 17 and has to be plotted separately for clarity of presentation. This is not the case
when a nonuniform grid is used for the interpolation as shown in the lower left panel of
ﬁgure 3.4. The interpolants approach the true function in the center and at the edges of
the interval. The points used in this case are the GaussLobatto roots of the Chebyshev
polynomial of degree N − 1, where N is the number of points. 46 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES 3.5 Compact Diﬀerencing Schemes A major disadvantage of the ﬁnite diﬀerence approach presented earlier is the widening
of the computational stencil as the order of the approximation is increased. These large
stencils are cumbersome near the edge of the domain where no data is available to
perform the diﬀerencing. Fortunately, it is possible to derive highorder ﬁnite diﬀerence
approximation with compact stencils at the expense of a small complication in their
evalution: implicit diﬀerencing schemes (as opposed to explicit schemes) must be used.
Here we show how these schemes can be derived. 3.5.1 Derivation of 3term compact schemes The Taylor series expansion of ui+m , where ui+m = u(xi + m∆x) about point xi can be
written as
∞
(m∆x)n (n)
u
(3.56)
ui+m =
n!
n=0
where u(n) is the nth derivative of u with respect to x at xi , with m being an arbitrary
number. From this expression it is easy to obtain the following sum and diﬀerence
ui+m ± ui−m =
ui+m + ui−m
2 = ui+m − ui−m
2m∆x = ∞ ((1 ± (−1)n ) n=0
∞ (m∆x)n (n)
u
n! (m∆x)n (n)
u
n!
n=0,2,4
∞ (m∆x)n (n+1)
u
(n + 1)!
n=0,2,4 (3.57)
(3.58)
(3.59) These expansion apply to arbitraty functions u as long as the expansion is valid; so they
apply in particular to their derivatives. In case we substitute u(1) for u in the summation
expansion we obtain an expression for the expansion of the derivatives:
(1) (1) ui+m + ui−m
2 = ∞ (m∆x)n (n+1)
u
n!
n=0,2,4 (3.60) Consider centered expansions of the following form
(1) (1) (1) αui−1 + ui + αui+1 = a1 δ2x ui + a2 δ4x ui + a3 δ6x ui (3.61) where α, and the ai are unknown constants. The Taylor series expansion of the left and
right hand sides can be matched as follows
(1) ui
or + 2α ∞ ∞
1
a1 + 2n a2 + 3n a3
∆xn u(n+1) =
∆xn u(n+1)
n!
(n + 1)!
n=0,2,4
n=0,2,4 ∞ (a1 + 2n a2 + 3n a3 ) − (n + 1)(δn0 + 2α)
∆xn u(n+1) = 0
(n + 1)!
n=0,2,4 (3.62) (3.63) 3.5. COMPACT DIFFERENCING SCHEMES 47 Here δn0 refers to the Kronecker delta: δnm = 0 for n = m and 1 if n = m. This leads to
the following constraints on the constants ai and α:
a1 + a2 + a3
n = 1 + 2α n a1 + 2 a2 + 3 a3 = 2(n + 1)α for n = 0 (3.64) for n = 2, 4, . . . , N (3.65) with a leading truncation error of the form:
a1 + 2N +2 a2 + 3N +2 a3 − 2(N + 3)α
∆xN +2 u(N +3)
(N + 3)! (3.66) Since we only have a handful of parameters we cannot hope to satisfy all these constraints
for all n. However, we can derive progressively better approximation by matching higherorder terms. Indeed with 4 paramters at our disposal we can only satisfy 4 constraints.
Let us explore some of the possible options. 3.5.2 Families of Fourth order schemes The smallest computational stencil is obtained by setting a2 = a3 = 0, in which case only
2 parameters are left to maximize the accuracy of the scheme, and only 2 constraints
can be imposed: a1 a1 − 2
3!
−2
2! α=1
α=0 with solution α a1 =
= 1
4
3
2 (3.67) A family of fourth order schemes can be obtained if we allow a wider stencil on the
right hand side of the equations, and allow a2 = 0. This family of schemes can be
generated by the single parameter α
a1 +
a2 = 1 + 2α
a1 + 4 a2 =
6α with solution a
1 a2 =
= 2
(α + 2)
3
1
(4α − 1)
3 (3.68) The leading terms in the truncation error would then be
TE =
= a1 + 24 a2 − 2 · 5α
a1 + 26 a2 − 2 · 7α
∆x4 u(5) +
∆x6 u(7)
5!
7!
4(3α − 1)
4(18α − 5)
∆x4 u(5) +
∆x6 u(7)
5!
7! (3.69)
(3.70) This family of compact scheme can be made unique by, for example, requiring the scheme
to be sixthorder and setting α = 1/3; the leading truncation error term would then be
4/(7!)∆x6 u(7) . 48 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES
10 0 10 −5 −5 10 ε 2 ε∞ 10 0 10 10 −10 −10 10 −15 −15 1 10 10 10 2 10 1 2 10 M M Figure 3.5: Convergence curves for the compact (solid ligns) and explicit centered diﬀerence (dashed lines) schemes for the sample function u = sin πx, x ≤ 1. The dashdot
lines refers to lines of slope 2,4,6, and 8. The left panel shows the 2norm of the error
while the right panel shows the ∞norm. 3.5.3 Families of Sixth order schemes Allowing the stencil on the right hand sides to expand, a3 = 0, we can generate families
of sixthorder schemes easily. The constraints are given by a1 a 1
a
1 +
a2 +
a3 = 1 + 2α
2a
2 a=
+2
6α
2+3
3
+ 24 a2 + 34 a3 =
10α with solution The leading terms in the truncation error would then be
TE =
= a1 a 2 a
3 =
=
= α+9
6
32α − 9
15
−3α + 1
10 (3.71) a1 + 28 a2 + 38 a3 − 2 · 9α
a1 + 26 a2 + 36 a3 − 2 · 7α
∆x6 u(7) +
∆x8 u(9)
(3.72)
7!
9!
12(−8α + 3)
72(−20α + 7)
∆x6 u(7) +
∆x8 u(9)
(3.73)
7!
9! Again, the formal order of the scheme can be made eighth order if one chooses α = 3/8. 3.5.4 Numerical experiments Figure 3.5 illustrate the convergence of the various compact scheme presented here for the
sample function u = sin πx. For comparison we have shown in dashed lines the convergence curves for the explicit ﬁnite diﬀerence schemes presented earlier. The dasheddot
lines are reference curves with slopes 2,4,6 and 8, respectively. For this inﬁnitly diﬀerentiable function, the various schemes achieve their theoretically expected convergence
order as the order fo the scheme is increased. It should be noted that, although, the
convergence curves of the two approaches are parallel (the same slope), the errors of the
compact schemes are lower then those of their explicit diﬀerene counterparts. Chapter 4 Application of Finite Diﬀerences
to ODE
In this chapter we explore the application of ﬁnite diﬀerences in the simplest setting
possible, namely where there is only one independent variable. The equations are then
referred to as ordinary diﬀerential equations (ODE). We will use the setting of ODE’s to
introduce several concepts and tools that will be useful in the numerical solution of partial
diﬀerential equations. Furthermore, timemarching schemes are almost universally reliant
on ﬁnite diﬀerence methods for discretization, and hence a study of ODE’s is time well
spent. 4.1 Introduction Here we derive how an ODE may be obtained in the process of solving numerically a
partial diﬀerential equations. Let us consider the problem of solving the following PDE:
ut + cux = νuxx , 0 ≤ x ≤ L (4.1) subject to periodic boundary conditions. Equation (4.1) is an advection diﬀusion equation with c being the advecting velocity and ν the viscosity coeﬀﬁcient. We will take
c and ν to be positive constants. The two independent variables are t for time and x
for space. Because of the periodicity, it is sensible to expand the unknown function in a
Fourier series:
∞
un (t)eikn x
ˆ u= (4.2) k =−∞ where un are the complex amplitudes and depend only on the time variable, whereas eikx
ˆ
are the Fourier functions with wavenumber kn . Because of the periodicity requirement
we have kn = 2πn/L where n is an integer. The Fourier functions form what is called an
orthonormal basis, and can be determined as follows: multiply the two sides of equations
(4.2) by e−ikm x where m is integer and integrate over the interval [0π ] to get:
L
0 ue−ikm x dx = ∞ L un (t)
ˆ k =−∞ 49 0 ei(kn −km )x dx (4.3) 50 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE Now notice that the integral on the right hand side of equation (4.3) satisﬁes the orthogonality property:
L i(kn −km )x e
0 dx = ei(kn −km )L − 1 i(kn − km ) L, n = m = ei2π(n−m)L − 1
= 0, n = m
i(kn − km ) (4.4) The role of the integration is to pick out the m − th Fourier component since all the other
integrals are zero. We end up with the following expression for the Fourier coeﬃcients:
um =
ˆ 1
L L
0 u(x)e−ikm x dx (4.5) Equation (4.5) would allow us to calculate the Fourier coeﬃcients for a known function
u. Note that for a real function, the Fourier coeﬃcients satisfy
u− n = u∗
ˆ
ˆn (4.6) where the ∗ superscript stands for the complex conjugate. Thus, only the positive Fourier
components need to be determined and the negative ones are simply the complex conjugates of the positive components.
The Fourier series (4.2) can now be diﬀerentiated term by term to get an expression
for the derivatives of u, namely:
ux =
uxx =
ut = ∞
k =−∞
∞
k =−∞
∞ ikn un (t)eikn x
ˆ (4.7) 2
ˆ
−kn un (t)eikn x (4.8) dˆn ikn x
u
e
dt
k =−∞ (4.9) Replacing these expressions for the derivative in the original equation and collecting
terms we arrive at the following equations:
∞
k =−∞ dun
ˆ
2
+ (ickn + νkn )ˆn eikn x = 0
u
dt (4.10) Note that the above equation has to be satisﬁed for all x and hence its Fourier amplitudes
must be zero for all x (just remember the orthogonality property (4.4), and replace u by
zero). Each Fourier component can be studied separately thanks to the linearity and the
constant coeﬃcients of the PDE.
The governing equation for the Fourier amplitude is now
dˆ
u
= −(ick + νk2 ) u
ˆ
dt
κ (4.11) 4.2. FORWARD EULER APPROXIMATION 51 where we have removed the subscript n to simplify the notation, and have introduced
the complex number κ. The solution to this simple ODE is:
u = u0 eκt
ˆˆ (4.12) where u0 = u(t = 0) is the Fourier amplitude at the initial time. Taking the ratio of the
ˆ
ˆ
solution between time t and t + ∆t, we can get see the expected behavior of the solution
between two consquetive times:
u(t + ∆t)
ˆ
= eκ∆t = eRe(κ)∆t eiI m(κ)∆t
u(t)
ˆ (4.13) where Re(κ) and I m(κ) refer to the real and imaginary parts of κ. It is now clear to
follow the evolution of the amplitude of the Fourier components:
u(t + ∆t) = u(t)eRe(κ)∆t
ˆ
ˆ (4.14) The analytical solution predicts an exponential decrease if Re(κ) < 0, an exponential
increase if Re(κ) > 0, and a constant amplitude if Re(κ) = 0. The imaginary part of κ
inﬂuences only the phase of the solution and decreases by an amount I m(κ)∆t. We now
turn to the issue of devising numerical solution to the ODE. 4.2 Forward Euler Approximation Let us approximate the time derivative in (4.11) by a forward diﬀerence approximation
(the name forward Euler is also used) to get:
un+1 − un
≈ κun
∆t (4.15) where the superscript indicates the time level:un = u(n∆t), and where we have removed
the ˆ for simplicity. Equation (4.15) is an explicit approximation to the original differential equation since no information about the unknown function at the future time
(n + 1)∆t has been used on the right hand side of the equation. In order to derive the
error committed in the approximation we rely again on Taylor series. Expanding un+1
about time level n∆ts, and inserting in the forward diﬀerence expression (4.15) we get:
ut − κu = ∆ t2
∆t
ut −
utt
2
3!
truncation error ∼O(∆t)
− (4.16) The terms on the right hand side are the truncation errors of the forward Euler approximation. The formal deﬁnition of the truncation error is that it is the diﬀerence
between the analytical and approximate representation of the diﬀerential equation. The
leading error term (for suﬃciently small ∆t) is linear in ∆t and hence we expect the
errors to decrease linearly. Most importantly, the approximation is consistent in that
the truncation error goes to zero as ∆t → 0. 52 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE
Given the initial condition u(t = 0) = u0 we can advance the solution in time to get:
u1 = (1 + κ∆t)u0
u2 = (1 + κ∆t)u1 = (1 + κ∆t)2 u0
u3 = (1 + κ∆t)u2 = (1 + κ∆t)3 u0
.
.
. (4.17) un = (1 + κ∆t)un−1 = (1 + κ∆t)n u0
Let us study what happens when we let ∆t → 0 for a ﬁxed time integration tn = n∆t.
The only factor we need to worry about is the numerical ampliﬁcation factor:(1 + κ∆t):
tn lim (1 + κ∆t) ∆t = lim e ∆t→0 tn ln(1+κ∆t)
∆t ∆t→0 = lim e tn (κ∆t−κ2 ∆t2 +...)
∆t ∆t→0 = eκtn (4.18) where we have used the logarithm Taylor series ln(1 + ǫ) = ǫ − ǫ2 + . . ., assuming that
κ∆t is small. Hence we have proven convergence of the numerical solution to the analytic
solution in the limit ∆t → 0. The question is what happens for ﬁnite ∆t?
Notice that in analogy to the analytic solution we can deﬁne an ampliﬁcation factor
associated with the numerical solution, namely:
un
= Aeiθ
(4.19)
un − 1
where θ is the argument of the complex number A. The amplitude of A will determine
whether the numerical solution is amplifying or decaying, and its argument will determine
the change in phase. The numerical ampliﬁcation factor should mimic the analytical
ampliﬁcation factor, and should lead to an anologous increase or decrease of the solution.
For small κ∆t it can be seen that A is just the ﬁrst term of the Taylor series expansion
of eκ∆t and is hence only ﬁrst order accurate. Let us investigate the magnitude of A in
terms of κ, a problem parameter, and ∆t the numerical parameter, we have:
A= A2 = AA∗ = 1 + 2Re(κ)∆t + κ2 ∆t2 (4.20) We focus in particular for the condition under which the amplitude factor is less then 1.
The following condition need then to be fullﬁlled (assuming ∆t > 0):
∆t ≤ −2 Re(κ)
κ2 (4.21) There are two cases to consider depending on the sign of Re(κ). If Re(κ) > 0 then A > 1
for ∆t > 0, and the ﬁnite diﬀerence solution will grow like the analytical solution. For
Re(κ) = 0, the solution will also grow in amplitude whereas the analytical solution
eκ
predicts a neutral ampliﬁcation. If Re(κ) < 0, then A > 1 for ∆t > −2 Rκ(2 ) whereas

the analytical solution predicts a decay. The moral of the story is that the numerical
solution can behave in unexpected ways. We can rewrite the ampliﬁcation factor in the
following form:
A2 = AA∗ = [Re(z ) + 1]2 + [I m(z )]2
(4.22) where z = κ∆t. The above equation can be interpreted as the equation for a circle
centered at (−1, 0) in the complex plane with radius A2 . Thus z must be within the
unit circle centered at (−1, 0) for A2 ≤ 1. 4.3. STABILITY, CONSISTENCY AND CONVERGENCE 4.3 53 Stability, Consistency and Convergence Let us denote by u the exact analytical solution to the diﬀerential equation, and by U
the numerical solution obtained using a ﬁnite diﬀerence scheme. The quantity U − u
is a norm of the error. we have the following deﬁnitions:
• Convergence A scheme is called to converge to O(∆tp ) if U − u = O(∆tp ) as
∆t → 0, where p is a positive constant.
• Truncation Error The local diﬀerence between the diﬀerence approximation and
the diﬀerential equations. It is the error introduced if the exact solution is plugged
into the diﬀerence equations.
• Consistency The ﬁnite diﬀerence approximation is called consistent with the differential equation if the truncation error goes to zero when the numerical parameters are made arbitrarily small.
• Stability A method is called stable if there is a constant C independent of the
time step or the number of time steps such that:
Un < C U0 (4.23) Equation (4.23) is a very loose constraint on the numerical approximation to guarantee convergence. This constraint allows the solution to grow in time (indeed
the solution can still grow exponentially fast), but rules out growth that depends
on the number of time steps or the step size. In situations where the analytical
solution is known not to grow, it is entirely reasonable to put the restriction:
Un < U0 4.3.1 (4.24) Lax Richtmeyer theorem The celebrated LaxRichtmeyer theorem links the notion of consistency and stability for
linear diﬀerential equations. It maintains that for linear diﬀerential equations, a consistent ﬁnite diﬀerence approximation converges to the true solution if the scheme is stable.
The converse is also true in that a convergent and consistent numerical solution must be
stable. We will show a here simpliﬁed version of the LaxRichtmeyer equivalence theorem
to highlight the relationships between consistency stability, and justify the constraints
we place on the ﬁnite diﬀerence approximations. A general form for the integration of
the ODE U takes the form:
U n = AU n−1 + bn−1
(4.25)
where A is the multiplicative factor and b a source sink term that does not depend on u.
If the exact solution is plugged into the above recursive formula we get:
un = Aun−1 + bn−1 + T n−1 ∆t (4.26) where T n−1 is the truncation error at time n. Substracting the two equations from
each others, and invoking the linearity of the process, we can derive an equation for the
evolution of the error in time, namely:
en = Aen−1 − T n−1 ∆t (4.27) 54 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE where en = U n − un is the total error at time tn = n∆t. The reapplication of this formula
to en−1 transforms it to:
en = A(Aen−2 − T n−2 ∆t) − T n−1 ∆t = (A)2 en−2 − ∆t AT n−2 + T n−1 (4.28) where (A)2 = A.A, and the remainder of the superscript indicate time levels. Repeated
application of this formula shows that:
en = (A)n e0 − ∆t (A)n−1 T 0 + (A)n−2 T 1 + . . . + AT n−2 + T n−1 (4.29) Equation (4.29) shows that the error at time level n depends on the initial error, on
the history of the truncation error, and on the discretization through the factor A. We
will now attempt to bound this error and show that this possible if the truncation error
can be made arbitrarily small (the consistency condition), and if the scheme is stable
according to the deﬁnition shown above. A simple application of the triangle inequality
shows that
en  ≤ An  e0  +∆t (A)n−1 T 0 + (A)n−2 T 1 + . . . + A T n−2 + T n−1 (4.30) Now we deﬁne T = max T m  for all 0 ≤ m ≤ n − 1, that is T is the maximum norm
of the truncation error encountered during the course of the calculation, then the right
hand side of the above inequality can be bounded again:
en  ≤ An  e0  + ∆tT n−1
m=0 (A)m  (4.31) In order to proceed further we need to introduce also the maximum bound on the
ampliﬁcation factor and all its powers. So let us assume that the scheme is stable, i.e.
there is a positive constant C , independent of ∆t and n, such that
max (Am ) ≤ C, for 0 ≤ m ≤ n (4.32) Since the individual entries in the summation are smaller then C and the sum involves n
terms, the sum must be smaller then nC . The inequality (4.32) is then bounded above
by C E 0  + nT ∆tC , and we arrive at:
en  ≤ C e0  + tn T C (4.33) where tn = n∆t is the ﬁnal integration time. The right hand side of (4.33) can be made
arbitrarily small by the following argument. First, the initial condition is known and so
the initial error is (save for roundoﬀ errors) zero. Second, since the scheme is consistent,
the maximum truncation error T can be made arbitrarily small by choosing smaller and
smaller ∆t. The end results is that the bound on the error can be made arbitrarily
small if the approximation is stable and consistent. Hence the scheme converges to
the solution as ∆t → 0. 4.4. BACKWARD DIFFERENCE 4.3.2 55 Von Neumann stability condition Here we derive a practical bound on the ampliﬁcation factor A based on the criteria
used in the derivation of the equivalence theorem Am  ≤ C :
Am  = Am ≤ C
A ≤ C (4.34)
1
m =e ∆t ln C
tm A ≤ 1 + O(∆t) (4.35)
(4.36) where we have used the Taylor series for the exponential in arriving at the ﬁnal expression.
This is the least restrictive condition on the ampliﬁcation factor that will permit us to
bound the error growth for ﬁnite times. Thus the modulus of the ampliﬁcation factor
maybe greater then 1 by an amount proportional to positive powers of ∆t. This gives
plenty of latitude for the numerical solution to grow, but will prevent this growth from
depending on the time step or the number of time steps.
In practice, the stability criterion is too generous, particularly when we know the
solution is bounded. The growth of the numerical solution should be bounded at all
times by setting C = 1. In this case the Von Neumann stability criterion reduces to
A ≤ 1 4.4 (4.37) Backward Diﬀerence The backward diﬀerence formula to the ODE is
du
dt tn+1 ≈ un+1 − un
= κun+1
∆t (4.38) This is an example of an implicit method since the unknown un+1 has been used in
evaluating the slope of the solution on the right hand side; this is not a problem to solve
for un+1 in this scalar and linear case. For more complicated situations like a nonlinear
right hand side or a system of equations, a nonlinear system of equations may have to
be inverted. It is easy to show via Taylor series analysis that the truncation error for
the backward diﬀerence scheme is O(∆t) and the scheme is hence consistent and of ﬁrst
order. The numerical solution can be updated according to:
un+1 = un
1 − κ∆t (4.39) and the ampliﬁcation factor is simply A = 1/(1 − κ∆t). Its magnitude is given by:
A2 = 1
1 − 2Re(κ)∆t + κ2 ∆t2 The condition under which this ampliﬁcation factor is bounded by 1 is (4.40) 56 4.5 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE Backward Diﬀerence The backward diﬀerence formula to the ODE is
du
dt ≈ tn+1 un+1 − un
= κun+1
∆t (4.41) This is an example of an implicit method since the unknown un+1 has been used in
evaluating the slope of the solution on the right hand side; this is not a problem to solve
for un+1 in this scalar and linear case. For more complicated situations like a nonlinear
right hand side or a system of equations, a nonlinear system of equations may have to
be inverted. It is easy to show via Taylor series analysis that the truncation error for
the backward diﬀerence scheme is O(∆t) and the scheme is hence consistent and of ﬁrst
order. The numerical solution can be updated according to:
un+1 = un
1 − κ∆t (4.42) and the ampliﬁcation factor is simply A = 1/(1 − κ∆t). Its magnitude is given by:
A2 = 1
1 − 2Re(κ)∆t + κ2 ∆t2 (4.43) The condition under which this ampliﬁcation factor is bounded by 1 is
∆t ≥ 2 Re(κ)
κ2 (4.44) again depend on the sign of Re(κ). If Re(κ) < 0, then A < 1 for all ∆t > 0; this
is an instance of unconditional stability. The numerical solution is also damped when
Re(κ) = 0 whereas the analytical solution is neutral. The numerical amplitude factor
can be rewritten as:
1
(4.45)
[1 − Re(z )]2 + [I m(z )]2 =
A2
and shows contours of constant amplitude factors to be circles centered at (1,0) and of
radius 1/A. 4.6 Trapezoidal Scheme The trapezoidal scheme is an an example of second order scheme that uses only two time
1
levels. It is based on applying the derivative at the intermediate time n + 2 , and using
a centered diﬀerence formula with step size ∆t/2. The derivation is as follows:
du
dt 1 = κun+ 2 (4.46) tn+ 1
2 un+1 + un
un+1 − un
+ O(∆t2 ) = κ
+ O(∆t2 )
∆t
2 (4.47) 4.6. TRAPEZOIDAL SCHEME 57 using simple Taylor series expansions about time n + 1 . The truncation error is O(∆t2 )
2
and the method is hence second order accurate. It is implicit since un+1 is used in the
evaluation of the right hand side. The unkown function can be updated as:
un+1 = 1+
1− κ ∆t
2
un
κ ∆t
2 (4.48) The ampliﬁcation factor is
1+
A=
1− κ ∆t
2
,
κ ∆t
2 2 A = 1 + Re(κ∆t) + 1 − Re(κ∆t) + κ2 ∆t2
4
κ2 ∆t2
4 (4.49) The condition for A < 1 is simply
Re(κ) ≤ 0 (4.50) The scheme is hence unconditionally stable for Re(∆t) < 0 and neutrally stable (A = 1)
for Re(κ) = 0.
Example 9 To illustrate the application of the diﬀerent scheme we proceed to evaluate
numerically the solution of ut = −iu with the initial condition u = 1. The analytical
solution in the complex u plane is a circle that starts at (1, 0) and proceeds counterclockwise. We then proceed to compute the numerical solutions using the forward, backward
and trapezoidal schemes.
The modulus of the forward Euler solution cycles outward and is indicative of the unstable nature of the scheme. The backward Euler solution cycles inward indicative of a
numerical solution that is too damped. Finally, the trapezoidal scheme has neutral ampliﬁcation: its solution remains on the unit circle and tracks the analytical solution quite
closely. Notice however, that the trapezoidal solution seem to lag behind the analytical
solution, and this lag seems to increase with time. This is symptomatic of lagging phase
errors. 4.6.1 Phase Errors Stability is primarily concerned with the modulus of the ampliﬁcation factor. However,
the accuracy of the numerical scheme depends on the amplitude and phase errors of the
scheme. The phase error can be analyzed by inspecting the argument of the ampliﬁcation
factor, the θ term in equation (4.19). The analytical change of phase for our model
problem is θe = I m(κ)∆t, the ratio of the numerical and analytical phase is called the
relative phase error:
θ
(4.51)
R=
θe
When R > 1 the numerical phase exceeds the analytical phase and we call the scheme
accelerating; if R < 1 the scheme is decelerating. For the forward diﬀerencing scheme
the relative phase is given by:
R= I m(z )
1
tan−1
I m(z )
1 + Re(z ) (4.52) 58 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE 2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2 −1 0 1 2 Figure 4.1: Solution of the oscillation equation using the forward (x), backward (+) and
trapezoidal schemes (◦). The analytical solution is indicated by a red asterisk.
1.3 RK3 1.2
RK4
1.1 Relative Phase 1
RK2
0.9
0.8
TZ
0.7
0.6
FD,BD
0.5
0.4
0 0.5 1 κ∆ t 1.5 2 2.5 Figure 4.2: Phase errors for several twolevel schemes when Re(κ) = 0. The forwad
and backward diﬀerencing schemes (FD and BD) have the same decelerating relative
phase. The Trapezoidal scheme (TZ) has lower phase error for the same κ∆t as the 2
ﬁrst order schemes. The Runge Kutta schemes of order 2 and 3 are accelerating. The
best performance is for RK4 which stays closest to the analytical curve for the largest
portion of the spectrum. 4.7. HIGHER ORDER METHODS 59 In general it is hard to get a simple formula for the phase error since the expressions often
involve the tan−1 functions with complicated arguments. Figure 4.2 shows the relative
phase as a function of κ∆t for the case where Re(κ) = 0 for several time integration
schemes. The solid black line (R = 1) is the reference for an exact phase. The forward,
backward and trapezoidal diﬀerencing have negative phase errors (and hence the schemes
are decelerating), while the RK schemes (to be presented below) have an accelerating
phase. 4.7
4.7.1 Higher Order Methods
Multi Stage (Runge Kutta) Methods One approach to increasing the order of the calculations without using information at
previous time levels is to introduce intermediate stages in the calculations. The most
popular of these approaches is referred to as the Runge Kutta methods. We will illustrate
their derivation for the second order scheme. For generality, we will assume that the ODE
takes the form
du
= f (u, t), u(0) = u0
(4.53)
dt
The derivation of the second order Runge Kutta method starts with the expression:
u(1) = un + a21 ∆t
u n+1 n (4.54)
n (1) = u + b1 ∆tf (u , tn ) + b2 ∆tf (u , tn + c2 ∆t) (4.55) where a21 , b1 , b2 and c2 are constant that will be determined on the basis of accuracy.
Variants of this approach has the constants determined on the basis of stability considertaion, but in the following we follow the accuracy criterion. The key to determining
these constants is to match the Taylor series expansion for the ODE with that of the
approximation. Expanding u as a Taylor series in time we get:
un+1 = un + ∆t du ∆t2 d2 u
+
+ O(∆t3 )
dt
2! dt2 (4.56) Notice that the ODE provides the information necessary to compute the derivative in
the Taylor series. Thus we have:
du
dt
d2 u
dt2
d3 u
dt3 = f (u, t) (4.57) ∂f
∂f du
df
=
+
dt
∂t
∂u dt
∂f
∂f
=
+
f
∂t
∂u
df
dft dfu
+
f + fu
=
dt
dt
dt
2
= ftt + fut f + fut f + fuu f 2 + fu ft + fu f
= 2 = ftt + 2fut f + fuu f + fu ft + 2
fu f (4.58)
(4.59)
(4.60)
(4.61)
(4.62) 60 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE Replacing these two derivatives in the Taylor series expression we get:
∆ t2
∆ t3
2
[fu f + ft ]tn +
[ftt +2fut f + fuuf 2 + fu ft + fu f ]+ O(∆t4 )
2!
3!
(4.63)
We now turn to expanding the expression for the proposed diﬀerence equations. We
have to proceed carefully to include the eﬀects of changes in u and t in our expansion.
We start by expanding the last term of equation (4.55) about the variable un . un+1 = un +∆tf (un , tn )+ f (un + a21 ∆tf, tn + c2 ∆t) = f (un , tn + c2 ∆t)+ a21 ∆tf fu + (a21 ∆tf )2
fuu + O(∆t3 ) (4.64)
2! Now each term in expansion (4.64) is expanded in the t variable about time tn to get:
(c2 ∆t)2
ftt + O(∆t2 )
2!
fu (un , tn + c2 ∆t) = fu (un , tn ) + fut (un , tn )c2 ∆t + O(∆t2 )
f (un , tn + c2 ∆t) = f (un , tn ) + c2 ∆tft +
n n fuu (u , tn + c2 ∆t) = fuu (u , tn ) + O(∆t) (4.65)
(4.66)
(4.67) Substituting these expressions in expansion (4.64) we get the twovariable Taylor series
expansion for f . The whole expression is then inserted in (4.55) to get:
un+1 = un + (b2 + b1 ) f ∆t + (b2 a21 f fu + b2 c2 ft ) ∆t2
+ b2 c2
2
2 ftt + b2 a21 c2 f fut + b2 a2
21
2 (4.68) fuu ∆t3 + O(∆t3 ) Matching the expansions (4.68) and (4.63) term by term we get the following equations
for the diﬀerent constants. b2 + b1 = 1 2b2 a21 = 1
(4.69) 2b c
=1
22 A solution can be found in term of the parameter b2 , and it is as follows: b1 a21 c
2 = 1 − b2
1
=
2b2
1
=
2b2 (4.70) A family of second order Runge Kutta schemes can be obtained by varying b2 . Two
common choices are
• Midpoint rule with b2 = 1, so that b1 = 0 and a21 = c2 =
becomes:
u(1) = un + ∆t
f (un , tn )
2 un+1 = un + ∆tf (u(1) , tn + 1
2. The schemes (4.71)
∆t
)
2 (4.72) The ﬁrst phase of the midpoint rule is a forward Euler half step, followed by a
centered approximation at the midtime level. 4.7. HIGHER ORDER METHODS
• Heum rule b2 = 1
2 61 and a21 = c2 = 1. u(1) = un + ∆tf (un , tn )
∆t
un+1 = un +
[f (u(1) , tn + ∆t) + f (un , tn )]
2 (4.73)
(4.74) The ﬁrst step is forward Euler full step followed by a centered step with the averaged
sloped.
Higher order Runge Kutta schemes can be derived by introducing additional stages
between the two time levels; their derivation is however very complicated (for more
information see Butcher (1987) and see Dormand (1996) for a more readable account).
Here we limit ourselves to listing the algorithms for common third and fourth order
Runge Kutta schemes.
• RK3 q1 = ∆tf (un , tn )
q2 = ∆tf (u(1) , tn +
q3 = ∆tf (u(2) , tn + ∆t
3 )−
∆t
3 )− 5q1
9
153q2
128 1
u(1) = un + q3
u(2) = u(1) + 15q2
16
un+1 = u(2) + 8q3
15 (4.75) • RK4 The fourth order RK scheme is the most wellknown for its accuracy and
large stability region. It is:
q1 =
q2 =
q3 =
q4 =
un+1 = 4.7.2 ∆tf (un , tn )
1
∆tf (un + q2 , tn + ∆t )
2
n + q 2 , t + ∆t )
∆tf (u
n
2
2
∆tf (un + q3 , tn + ∆t)
un + q1 +2q2 +2q3 +q4
6 (4.76) Remarks on RK schemes The following is a quick summary of the RK properties of diﬀerent orders.
1. Implicit Runge Kutta time steps are possible.
2. Runge Kutta oﬀers high order integration with only information at two time levels.
Automatic step size control is easy since the order of the method does not depend
on maintaining the same step size as the calculation proceeds. Several strategies
are then possible to maximize accuracy and reduce CPU cost.
3. For order less then or equal to 4, only one stage is required per additional order,
which is optimal. For higher order, it is known that the number of stages exceeds
the order of the method. For example, a ﬁfth order method requires 6 stages, and
an eight order RK scheme requires a minimum of 11 stages.
4. Runge Kutta schemes require multiple evaluation of the right hand side per time
step. This can be quite costly if a large system of simultaneous ODE’s is involved.
Alternatives to the RK steps are multilevel schemes. 62 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE
3.5 RK4 3 2.5 1 RK3 AB2 Im(κ∆ t) 0.8 AB3 0.6 2
RK2 0.4
0.2 1.5 0 RK1 1 −0.2
−0.4
−0.6 0.5 −0.8 0
−3 −1 −2.5 −2 −1.5
−1
Re(κ∆ t) −0.5 0 0.5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 Figure 4.3: Stability region for the RungeKutta methods of order 1, 2, 3, and 4 (left
ﬁgure), and for the Adams Bashforth schmes of order 2 and 3 (right ﬁgure). The RK2
and AB2 stability curves are tangent to the imaginary axis at the origin, and hence the
method are not stable for purely imaginary κ∆t.
5. A new family of Runge Kutta scheme was devised in recent years to cope with the
requirements of Total Variations Diminishing (TVD) schemes. For second order
methods, the Heum scheme is TVD. The third order TVD Runge Kutta scheme is
1
u(1) = un + q3
3 n 1 (1) 1
q2 = ∆tf (u(1) , tn + ∆t) u(2) = u + u + q2
4
4
4
(2) , t + ∆t ) un+1 = 1 un + 2 u(2) + 2 q
q3 = ∆tf (u
n
3
2
3
3
3 q1 = ∆tf (un , tn ) 4.7.3 (4.77) Multi Time Levels Methods The RungeKutta methods achieve their accuracy by evaluating the right hand side
function at intermediate time levels. The cost of this accuracy is a multiple evaluation
of f for a an integration of size ∆t. This may prove to be expensive if we are looking
at complicated right hand sides and/or systems of ODEs. An alternative is to use
information prior to tn to increase the order of accuracy at tn+1 .
Leap Frog scheme
The leap frog scheme consists of using a centered diﬀerence in time at level n:
un+1 = un−1 + 2∆tf (un , tn ) (4.78) It is easy to show that the truncation error is of size O(∆t2 ). Moreover, unlike the trapezoidal scheme, it is explicit in the unknown un+1 , and hence does not involve nonlinear 4.7. HIGHER ORDER METHODS 63 complications, nor systems of equations. For our model equation, the trapezoidal scheme
takes the form:
un+1 = un−1 + 2κ∆tun
(4.79)
The determination of the ampliﬁcation factor is complicated by the fact that two time
levels are involved in the calculations. Nevertheless, let us assume that the ampliﬁcation
factor is the same for each time step, i.e. un = Aun−1 , and un+1 = Aun . We then arrive
at the following equation:
A2 − 2zA − 1 = 0
(4.80)
There are two solutions to this quadratic equation:
A± = z ± 1 + z2 (4.81) In the limit of good resolution, z  → 0, we have A+ → 1 and A− → −1. The numerical
solution is capable of behaving in two diﬀerent ways, or modes. The mode associated
with A+ is referred to as the physical mode because it approximates the solution to
the original diﬀerential equation. The mode associated with A− is referred to as the
computational mode since it arises solely as an artifact of the numerical procedure.
The origin of the computational mode can be traced back to the fact that the leapfrog scheme is an approximation to a higher order equation that requires more initial
conditions then necessary for the original ODE. To show this consider the trivial case
of κ = 0 where the analytical solution is simply given by ua = u0 ; here u0 is the initial
condition. The amplitude factors for the leapfrog schemes are A+ = 1 and A− = −1,
and hence the computational mode is expected to keep its amplitude but switch sign at
every time step. Applying the leapfrog scheme we see that all even time levels will have
the correct value: u2 = u4 = . . . = u0 . The odd time levels will be contaminated by
error in estimating the second initial condition needed to jump start the calculations. If
u1 = u0 + ǫ where ǫ is the initial error committed, the solution at all odd time levels
will then be u2n+1 = u0 + ǫ. The numerical solution for the present simple case can be
written entirely in terms of the physical (initial condition) and computational (initial
condition error) modes:
ǫ
ǫ
(4.82)
un = u0 + − (−1)n
2
2
Absolute stability requires that A± ≤ 1; notice however that the product of the
two roots is A+ A− = −1, which implies that A+ A−  = 1. Hence, if one root, say
A+ is stable A+  < 1, the other one must be unstable with A−  = 1/A+  > 1; the
only exception is when both ampliﬁcation factor have a neutral ampliﬁcation A+  =
A−  = 1. For real z , Im(z ) = 0, one of the two roots has modulus exceeding 1, and
the scheme is always unstable. Let us for a moment assume that z = iλ , we then have:
√
A = iλ + 1 − λ2 . If λ ≤ 1 then the quantity under the square root sign is positive and
we have two roots such that A+  = A−  = 1. To make further progress on studying the
stability of the leap frog scheme, let z = sinh(w) where w is a complex number. Using
the identity cosh2 w − sinh2 w = 1 we arrive at the expression A± = sinh w ± cosh w.
Setting w = a + ib where a, b are real, subsituting in the previous expression for the
ampliﬁcation factor, and calculating its modulus we get: A±  = e±a . Hence a = 0 for 64 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE both ampliﬁcation factors to be stable. The region of stability is hence z = i sin b where
b is real, and is conﬁned to the unit slit along the imaginary axis I m(z ) ≤ 1.
The leap frog scheme is a popular scheme to integrate PDE’s of primarily hyperbolic
type in spite of the existence of the computational mode. The reason lies primarily in
its neutral stability and good phase properties. The control of the computational mode
can be eﬀectively achieved either with a Asselin time ﬁlter (see Durran (1999)) or by
discarding periodically the solution at level n − 1 and taking a two time level scheme.
MultiStep schemes
A family of multistep schemes can built upon interpolating the right hand side of the
ODE in the interval [tn tn+1 ] and performing the integral. The derivation starts from the
exact solution to the ODE:
un+1 = un + tn+1 f (u, t) dt (4.83) tn Since the integrand is unknown in [tn tn+1 ] we need to ﬁnd a way to approximate it given
information at speciﬁc time levels. A simple way to achieve this is to use a polynomial
that interpolates the integrand at the points (tk , uk ), n − p ≤ k ≤ n, where the solution
is known. If we write:
f (u, t) = Vp (t) + Ep (t)
(4.84)
where Vp is the polynomial approximation and Ep the error associated with it, then the
numerical scheme becomes:
un+1 = un + tn+1 Vp (t) dt + tn tn+1
tn Ep (t) dt (4.85) If the integration of Vp is performed exactly, then the only approximation errors present
are due to the integration of the interpolation error term; this term can be bounded by
max(Ep ∆t.
The explicit family of Adams Bashforth scheme relies on Lagrange interpolation.
Speciﬁcally,
p hp (t)f n−p ,
k Vp (t) = (4.86) k =0
p hp (t) =
k
m=0,m=k = t − t n−m
t n−k − t n−m (4.87) t − tn−(k−1)
t − t n−k −1
t − tn−p
t − tn
...
...
tn−k − t n
tn−k − tn−(k−1) tn−k − tn−k−1
t n−k − tn−p (4.88) It is easy to verify that hp (t) is a polynomial of degree p − 1 in t, and that hp (tn−m ) = 0
k
k
for m = k and hp (tn−k ) = 1. These last two properties ensures that Vp (tn−k ) = f n−k .
k
The error associated with the Lagrange interpolation with p + 1 points is O(∆tp+1 ).
Inserting the expressions for Vp in the numerical scheme, and we get:
p un+1 = un + f n−k
k =0 tn+1
tn hp (t) dt
k + ∆tO(∆tp+2 ) (4.89) 4.7. HIGHER ORDER METHODS 65 Note that the error appearing in the above formula is only the local error, the global
error is one order less, i.e. it is O(∆tp+1 ).
We illustrate the application of this procedure by considering the derivation of its
second and third order variants. The second order scheme requires p = 1. Hence, we
write:
t − tn
t − tn−1 n
f+
f n−1
(4.90)
V1 (t) =
tn − t n−1
tn−1 − tn The integral on the interval [tn tn+1 ] is
tn+1
tn tn+1 t − tn−1
dtf n +
t n − t n−1
tn
3 n 1 n−1
= ∆t
f−f
2
2 V1 (t) dt = tn+1
tn t − tn
dtf n−1
t n−1 − t n (4.91)
(4.92) The ﬁnal expression for the second order Adams Bashforth formula is:
un+1 = un + ∆t 3 n 1 n−1
f−f
+ O(∆t3 )
2
2 (4.93) A third order formula can be designed similarly. Starting with the quadratic interpolation polynomial V2 (t):
V2 (t) =
+ [t − tn ][t − tn−2 ]
[t − tn−1 ][t − tn−2 ] n
f+
f n−1
[tn − tn−1 ][tn − tn−2 ]
[tn−1 − tn ][tn−1 − tn−2 ]
[t − tn ][t − tn−1 ]
f n−2
[tn−2 − tn ][tn−2 − tn−1 ] (4.94) Its integral can be evaluated and plugged into equation (4.89) to get:
un+1 = un + ∆t 23 n 16 n−1
5
f− f
+ f n−2
12
12
12 (4.95) The stability of the AB2 scheme can be easily determined for the sample problem.
The ampliﬁcation factors are the roots to the equation
z
3
A2 − 1 + z A + = 0
2
2 (4.96) and the two roots are: 1
3
A± = 1 + z ±
2
2 3
1+ z
2 2 − 2z . (4.97) Like the leap frog scheme AB2 suﬀers from the existence of a computational mode. In the
limit of good resolution, z → 0, we have A+ → 1 and A+ → 0; that is the computational
mode is heavily damped. ﬁgure 4.4 shows the modulus of the physical and computational
modes for Re(z ) = 0 and Im(z ) < 1. The modulus of the computational mode amplitude
factor is quite small for the entire range of z considered. On the other hand the physical
mode is unstable for purely imaginary z as the modulus of its ampliﬁcation factor exceeds 66 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE
1.6
1.4
1.2 A± 1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 κ∆ t 0.6 0.8 1 Figure 4.4: Modulus of ampliﬁcation factor for the physical and computational modes
of AB2 when Re(κ) = 0.
1. Note however, that a series expansion of A+  for small z = iλ∆t shows that A+  =
1 + (λ∆t4 )/4 and hence the instability grows very slowly for suﬃciently small ∆t. It
can be anticipated that AB3 will have one physical mode and two computational modes
since its stability analysis leads to a third order equation for the ampliﬁcation factor.
Like AB2, AB3 strongly damps the two computational modes; it has the added beneﬁt
of providing conditional stability for Im(z ) = 0. The complete stability regions for AB2
and AB3 is shown in the right panel of ﬁgure 4.3.
Like all multilevel schemes there are some disadvantages to Adams Bashforth methods. First a starting method is required to jump start the calculations. Second the
stability region shrinks with the order of the method. The good news is that although
AB2 is unstable for imaginary κ, its instability is small and tolerable for ﬁnite integration
time. The third order Adams Bashforth scheme on the other hand includes portion of
the imaginary axis, which makes AB3 quite valuable for the integration of advection like
operators. The main advantage of AB schemes over RungeKutta is that they require
but one evaluation of the right hand side per time step and use a similar amount of
storage. 4.8 Strongly Stable Schemes Occasionally we are interested in enlarging the stability region as much as possible,
while maitaining a high convergence order. The lowest order scheme of that sort is the
backward diﬀerence formula. We now look for equivalent higher order formula. The
common thread is to evaluate the derivative term at the next time level. The taylor
series expansion of un−k about time level un+1 is:
un−k+1 = un+1 − (k∆t) du (k∆t)2 d2 u (k∆t)3 d3 u (k∆t)4 d4 u
+
−
+
+ ...
1! dt
2! dt2
3! dt3
4! dt4 (4.98) 4.8. STRONGLY STABLE SCHEMES 67 p
k =1 kak p
a1
a2
a3
a4
1
1
2
4/3
−1/3
3 18/11 −9/11 2/11
4 48/25 −36/25 16/25 −3/25 1
2/3
6/11
12/25 Table 4.1: Coeﬃcients of the Backward Diﬀerence Formula of order 1, 2, 3 and 4.
where k = 1, . . . , p. Multiplying each of these expansion by a coeﬃcient ak and adding
the individual terms, we get:
p p p
n−k +1 ak u n+1 ak u =
k =1
p k =1 −
− − (k∆t)3 k3 ak 3! k =1
p km ak
k =1 kak
k =1
3 du
+
dt3 p (k∆t) du
+
1! dt k2 ak k =1
(k∆t)2 d4 u p k4 ak 4! k =1 dt4 (k∆t)m dm u
+ ...
m! dtm (k∆t)2 d2 u
2! dt2 + ...
(4.99) For a pth order expression, we require the higher order derivative, 2 through p, to vanish;
this yields p − 1 homogeneous algebraic equations. For a nontrivial solution we need
to append one more conditions which we choose to be that the sum of the unknown
coeﬃcient is equal to one. This yields the following system of equations
p ak = 1 (4.100) k =1
p km ak = 0, m = 2, 3, . . . , p (4.101) k =1 for the p coeﬃcients a1 , a2 , ..., ap . In matrix form we have 111
1 22 32
1 23 33
..
.
..
.
..
.
p 3p
12 ... 1
a1
. . . p2 a2 . . . p3 a3 = .
. . .
. . .
.
. p
... p
ap 1
0
0
.
.
.
0 (4.102) The solution of this system for p equal to 2, 3 and 4 is shown in table 4.1. The
corresponding expressions are:
un+1 = un + ∆t ut n+1 + O(∆t)
4 n 1 n−1 2
u−u
+ ∆t ut n+1 + O(∆t2 )
un+1 =
3
3
3
9 n−1
2
6
18 n
u− u
+ un−2 + ∆t ut n+1 + O(∆t3 )
un+1 =
11
11
11
11 (4.103)
(4.104)
(4.105) 68 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE
4
3.5
3
2.5
2
1.5
1
0.5
0
−2 0 BDF1
2 BDF2
4 BDF3
6 8 10 Figure 4.5: Stability regions for the Backward Diﬀerencing schemes of order 1, 2 and 3.
The schemes are unstable within the enclosed region and stable everywhere else. The
instability regions grow with the order. The stability regions are symmetric about the
real axis
un+1 = 48 n 36 n−1 16 n−2
3
12
u− u
+u
− un−3 + + ∆t ut n+1 + O(∆t4 )(4.106)
25
25
25
25
25 Notice that we have shown explicitly the time level at which the time derivative is
approximated. The BDF’s scheme lead to implicit expressions to update the solution at
time level un+1 . Like the AdamsBashforth formula the BDF schemes require a starting
method. They also generate computational modes whose number depends on how many
previous time levels have been used. Their most important advantage is their stability
regions in the complex plane which are much larger then equivalent explicit schemes. 4.8.1 Stability of BDF We investigate the stability of the BDF schemes for the simple case where f (u, t) = κu.
It has already been shown that the backward diﬀerence scheme is stable in the entire
complex plane save for the inside of the unit circle centered at (1, 0). The equation for
the BDF2 ampliﬁcation factor is easily derived:
4
1
2
1 − z A2 − A + = 0
3
3
3
and admits the two roots: √
2 ± 1 + 2z
A± =
3 − 2z (4.107) (4.108) The positive roots is the physical mode while the negative root is the computational
mode. In the limit of z → 0, we have A+ → 1 and A− → 1/3, the computational mode
is hence naturally damped. Figure 4.5 shows the stability regions for the BDF schemes.
The contours of A = 1 are shown in the ﬁgure. The schemes are unstable within the
regions shown and stable outside it. The instability region grows with increasing order. 4.9. SYSTEMS OF ODES 4.9 69 Systems of ODEs The equations to be solved can form a system of equations:
du
= Lu
dt (4.109) where now u represents a vector of unknown and L is a matrix. The preceding schemes
can be all made to work with the system of equations by treating all the components in a
consistent matter. The major problem is not computational per se, but conceptual and
concerns the stability of a system of equation. For example, a backward diﬀerentiation
of the system leads to the following set of equations for the unknowns at the next time
level:
un+1 − un
= Lun+1 , or un+1 = P un , P = I − ∆tL
(4.110)
∆t
If we denote the exact solution of the system as v, then the error between the numerical
and exact solutions is given by en = un − v(tn ). A number of vector norms are usefull to
measure the error, namely, the 1, 2 or inf ty norms. The numerical solution converges
if e → 0 as ∆t → 0. The concept of stability also carries over. It is clear that the
solution, for a linear system at least, evolves as un = P n u0 . And hence the solution will
remain bounded if
un = P n u0 ≤ C u0
(4.111)
We now have to worry about the eﬀect of the ampliﬁcation matrix P . The problem
with the stability analysis is that it is hard to relate P n and P . The matrix norms
guarantee that P n ≤ P n . Hence requiring that P < 1 will ensure stability. This
is a suﬃcient condition but not a necessary condition. Since the spectral radius is a
lower bound on the diﬀerent matrix norms, it is necessary to require ρ(P ) ≤ 1. If P
can be made diagonal, such as when it possess a complete set of linearly independent
eigenvectors, then the requirement ρ(P ) ≤ 1 is suﬃcient and necessary. 70 CHAPTER 4. APPLICATION OF FINITE DIFFERENCES TO ODE Chapter 5 Numerical Solution of PDE’s
5.1 Introduction Suppose we are given a wellposed problem that consists of a partial diﬀerential equation
∂u
= Lu
∂t (5.1) where L is a diﬀerential operator, initial conditions
u(x, 0) = u0 (x) (5.2) and appropriate boundary conditions. We are now interested in devising a numerical
scheme based on ﬁnite diﬀerence method to solve the above wellposed problem.
Let v (x, t) be the exact solution of the problem. The numerical solution of these
equations via ﬁnite diﬀerences requires us to replace the continuous derivatives by discrete
approximations, and to conﬁne ourselves with the solution of the problem at a discrete set
of space and time points. Hence the numerical solution, denoted by u will be determined
at the discrete space points xj = j ∆x, and time points n∆t. We will use the notation
un = u(xj , tn ). The approximation must be consistent, stable, and convergent to
j
be useful in modeling physical problems. We will turn to the issue of deﬁning these
important concepts shortly.
A simple example of this class of problem is the scalar advection equation in a single
space dimension
ut + cux = 0,
(5.3)
where c is the advection speed. In this case L = −cux , and an appropriate boundary
conditions is to specify the value of u at the upstream boundary. To make the discussion
more concrete let us illustrate the discretization process for the case mentioned above.
For simplicity we assume that c is constant and positive. A simple ﬁnite diﬀerence scheme
that would advance the solution in time for time level n to n +1 is to use a Forward Euler
scheme for the time derivative, and a backward Euler scheme for the space derivative.
We get the following approximation to the PDE at point (xj , tn ).
un+1 − un
un − un−1
j
j
j
j
+c
=0
∆t
∆x
71 (5.4) 72 CHAPTER 5. NUMERICAL SOLUTION OF PDE’S Equation 5.4 provides a simple formula for updating the solution at time level n + 1 from
the values at time n:
un+1 = (1 − µ)un + µun−1 , where µ =
j
j
j c∆t
∆x (5.5) The variable µ is known as the Courant number and will ﬁgure prominently in the study
of the stability of the scheme. Equation 5.5 can be written as a matrix operation in the
following form: u1
u2
.
.
. uj −1 uj u j +1 . .
. uN −1 uN n+1 1 µ 1−µ ..
.. .
. = µ 1−µ
µ
1−µ
µ
1−µ
..
..
.
.
µ 1−µ
µ
1−µ u1
u2
.
.
. uj −1 uj u j +1 . .
. uN −1 uN
(5.6) where we have assumed that the boundary condition is given by u(x1 , t) = u0 (x1 ).
The following legitimate question can now be asked:
1. Consistency: Is the discrete equation (5.4) a correct approximation to the continuous form, eq. (5.3), and does this discrete form reduce to the PDE in the limit
of ∆t, ∆x → 0.
n
2. Convergence Does the numerical solution un → vj as ∆t, ∆x → 0.
j 3. Errors What are the errors committed by the approximation, and how should one
expect them to behave as the numerical resolution is increased.
4. Stability Does the numerical solution remained bounded by the data specifying
the problem? or are the numerical errors increasing as the computations are carried
out.
We will now turn to the issue of deﬁning these concepts more precisely, and hint to
the role they play in devising ﬁnite diﬀerence schemes. We will return to the issue of
illustrating their applications in practical situations later. 5.1.1 Convergence n
Let en = un − vj denote the error between the numerical and analytical solutions of the
j
j
PDE at time n∆t and point j ∆x. If this error tends to 0 as the grid and time steps are
decreased, the ﬁnite diﬀerence solution converges to the analytical solution. Moreover,
a ﬁnite diﬀerence scheme is said to be convergent of order (p, q ) if e = O(∆tp , ∆xq ) as
∆t, ∆x → 0. n 5.2. TRUNCATION ERROR 5.1.2 73 Truncation Error If the analytical solution is inserted in the ﬁnite diﬀerence scheme, we expect a small
residual to remain. This residual characterizes the error in approximating the continuous form by a discrete form. By performing a Taylor series analysis we can derive an
expression of this residual in terms of higher order derivatives of the solution. 5.1.3 Consistency Loosely speaking the notion of consistency addresses the problem of whether the ﬁnite
diﬀerence approximation is really representing the partial diﬀerential equations. We say
that a ﬁnite diﬀerence approximation is consistent with a diﬀerential equation if the
FD equations converge to the original equations as the time and space grids are reﬁned.
Hence, if the truncation error goes to zero as the time and space grids are reﬁned we
conclude that the scheme is consistent. 5.1.4 Stability The notion of stability is a little more complicated to deﬁne. Our primary concern here
is to make sure that numerical errors do not swamp the analytical solution. One way
to ensure that is to require the solution to remain bounded by the initial data. Hence a
deﬁnition of stability is to require the following
un ≤ C u0 (5.7) where C is a positive constant that may depend on the ﬁnal integration time tn but not
on the time nor on the space increments. Notice that this deﬁnition of stability is very
general and does not refer to the behavior of the continuum equation. If the latter is
known to preserve the norm of the solution, then the more restrictive condition
un ≤ u0 (5.8) is more practical, particularly for ﬁnite ∆t. 5.1.5 LaxRichtmeyer Equivalence theorem The LaxRichtmeyer equivalence theorem ties these diﬀerent notions together. It states
the following ”Given a properlyposed linear initial value problem, and a ﬁnite diﬀerence
approximation to it that satisﬁes the consistency condition, stability is the necessary and
suﬃcient condition for convergence. This theorem’s value is that it guarantees convergence provided two simpler conditions are satisﬁed, namely consistency and stability.
These two are considerable easier to check for general type problems then convergence. 5.2 Truncation Error The analysis of the truncation error for the simple advection equation will be presented
here. Inserting the exact solution v in the ﬁnite diﬀerence form 5.4, we get:
n
n
n
n
vj +1 − vj
vj − vj −1
+c
=0
∆t
∆x (5.9) 74 CHAPTER 5. NUMERICAL SOLUTION OF PDE’S The Taylor series, in time and space yield the following:
∆ t2
vtt n + O(∆t3 )
j
2
∆ x2
n
vxx n + O(∆x3 )
= vj − ∆x vx n +
j
j
2 n
n
vj +1 = vj + ∆t vt n +
j (5.10) n
vj −1 (5.11) Substituting these expressions in equation 5.9 we get:
[vt + cvx ] = − c∆x
∆t
vtt +
vxx + O(∆t2 , ∆x2 )
2
2 (5.12) T.E.
The terms on the left hand side of eq. 5.12 are the original PDE; all terms on the
right hand side are part of the truncation error. They represent the residual by which
the exact solution fails to satisfy the diﬀerence equation. For suﬃciently small ∆t and
∆x, the leading term in the truncation series is linear in both ∆t and ∆x. Notice also
that one can regard equation 5.12 as the true partial diﬀerential equation represented by
the ﬁnite diﬀerence equation for ﬁnite ∆t and ∆x. The analysis of the diﬀerent terms
appearing in the truncation error series can give valuable insight into the behavior of the
numerical approximation, and forms the basis of the modiﬁed equation analysis. We
will return to this issue later. For now it is suﬃcient to notice that the truncation error
tends to 0 as ∆t, ∆x → 0, and hence the ﬁnite diﬀerence approximation is consistent. 5.3 The Lax Richtmeyer theorem A general formula for the evolution of the ﬁnite diﬀerence solution is the following:
un = Aun−1 + bn−1 (5.13) where A is the evolution matrix, and b is a vector containing forcing terms and the
eﬀects of boundary conditions. The vector un holds the vector of solution values at time
n. The truncation error at a speciﬁc time level can be obtained by applying the above
matrix operation to the vector of exact solution values:
vn = Avn−1 + bn−1 + zn−1 ∆t (5.14) where z is the vector of truncation error at time level n − 1. Substracting equation 5.14
from 5.13, we get an evolution equation for the error, namely:
en = Aen−1 + zn−1 ∆t (5.15) Equation 5.15 shows that the error at time level n is made up of two parts. The ﬁrst
one is the evolution of the error inherited from the previous time level, the ﬁrst term on
the right hand side of eq. 5.15, and the second part is the truncation error committed
at the present time level. Since, this expression applies to a generic time level, the same
expression holds for en−1 :
en−1 = Aen−2 + zn−2 ∆t
(5.16) 5.3. THE LAX RICHTMEYER THEOREM 75 where we have assumed that the matrix A does not change with time to simplify the discussion (this is tantamount to assuming constant coeﬃcients for the PDE). By repeated
application of this argument we get:
en = A2 en−2 + Azn−2 + zn−1 ∆t (5.17) = A3 en−3 + A2 zn−3 + Azn−2 + zn−1 ∆t (5.18) .
.
.
= An e0 + An z0 + An−1 z1 + . . . + Azn−2 + zn−1 ∆t (5.19) Equation 5.19 shows that the error growth depends on the truncation error at all time
levels, and on the discretization through the matrix A. We can use the triangle inequality
to get a bound on the norm of the error. Thus,
e n ≤ An e0 + An z0 + An−1 z1 + . . . + A zn−2 + zn−1 ∆t
(5.20)
In order to make further progress we assume that the norm of the truncation error at
any time is bounded by a constant ǫ such that
ǫ= max 0≤m≤n−1 ( zm ) (5.21) The right hand side of inequality 5.20 can be bounded by
e n ≤ An e0 + ǫ∆ t n−1 Am (5.22) m=0 The initial errors and the subsequent truncation errors are thus modulated by the evolution matrices Am . In order to prevent the unbounded growth of the error norm as
n → ∞, we need to put a limit on the norm of the these matrices. This is in eﬀect the
stability property needed for convergence:
Am ≤ C = max ( Am )
1≤m≤n (5.23) where C is a constant independent of n, ∆t and ∆x. The sum in bracket can be bounded
by the factor nC ; the ﬁnal expression becomes:
en ≤ C e0 + t n ǫ (5.24) where tn = n∆t is the ﬁnal integration time. When ∆x → 0, the initial error en can be
made as small as desired. Furthermore, by consistency, the truncation error ǫ → 0 when
∆t, ∆x → 0. The global error is hence guarateed to go to zero as the computational grid
is reﬁned, and the scheme is convergent. 76 CHAPTER 5. NUMERICAL SOLUTION OF PDE’S 5.4 The Von Neumann Stability Condition The sole requirements we have put on the scheme for convergence are consistency and
stability. The latter took the form:
Am ≤ C (5.25) where C is independent of ∆t, ∆x and n. By the matrix norm properties we have:
Am ≤ A
hence it is suﬃcient to require that A
1 m ∆t m (5.26) ≤ C , or that A ≤ C m = e tm ln C = 1 + ln C
∆t + . . . = 1 + O(∆t)
tm (5.27) The Von neumann stability condition is hence that
A ≤ 1 + O(∆t) (5.28) Note that this stability condition does not make any reference on whether the continuous (exact) solution grows or decays in time. Furthermore, the stability condition is
established for ﬁnite integration times with the limit ∆t → 0. In practical computations
the computations are necessarily carried out with a small but ﬁnite ∆t, and it is frequently the case that the evolution equation puts a bound on the growth of the solution.
Since the numerical solution and its errors are subject to the same growth factors via the
matrix A, it is reasonable, and in most cases essential to require the stronger condition
A ≤ 1 for stability for non growing solutions.
A ﬁnal practical detail still needs to be ironed out, namely what norm should be
used to measure the error? From the properties of the matrix norm it is immediately
clear that the spectral radius ρ(A) ≤ A , hence ρ(A) ≤ 1 is a necessary condition
for stability but not suﬃcient. There are classes of matrices A where it is suﬃcient,
for example those that posess a complete set of linear eigenvectors such as those that
arise from the discretization of hyperbolic equation. If the 1 or ∞norms are used the
condition for stability becomes suﬃcient.
Example 10 In the case of the advection equation, the matrix A given in equation 5.6
has norm:
A 1 = A ∞ = µ + 1 − µ
(5.29)
For stability we thus require that µ + 1 − µ ≤ 1. Two cases need to be considered:
1. 0 ≤ µ ≤ 1: A = µ + 1 − µ = 1, stable.
2. µ < 0: A = 1 − 2µ > 1, unstable.
3. µ > 1: A = 1 + 2µ > 1, unstable.
The scheme is hence guaranteed to converge when 0 ≤ µ ≤ 1. 5.5. VON NEUMANN STABILITY ANALYSIS 5.5 77 Von Neumann Stability Analysis Matrix analysis can be diﬃcult to carry out for complicated PDEs, particularly since
it requires us to know the entire spectrum, or norm of the matrix before we can derive
useful stability criteria. Von Neumann devised a substantially easier stability test, one
that does involve matrices per se but can be reduced to evaluating scalars. The idea is to
restrict attention to periodic problems and to consider the Fourier modes of the solution.
Since the solution is periodic it can be expanded into a Fourier series of the form:
un = un eikxj
ˆk
j (5.30) where k is the wavenumber and un is its (complex) Fourier amplitude. This expression
ˆk
can then be inserted back in the ﬁnite diﬀerence equation, and an expression for the
ampliﬁcation factor A can be obtained, where A depends on k, ∆x and ∆t. Stability of
every Fourier mode will guarantee the stability of the entire solution, and hence A ≤ 1
for all Fourier modes is the necessary and suﬃcient stability condition for nongrowing
solutions.
Example 11 Inserting the Fourier series in the ﬁnite diﬀerence approximation for the
advection equation we end up with the following equation:
uk
ˆk
un+1 eikxj = (1 − µ)ˆn eikxj + µun eikxj−1
ˆk (5.31) Since xj −1 = xj − ∆x, the exponential factor drops out of the picture and we end up
with the following expression for the growth of the Fourier coeﬃcients:
ˆk
un+1 = (1 − µ) + µe−ik∆x un
ˆk (5.32) A The expression in bracket is nothing but the ampliﬁcation factor for Fourier mode k.
Stability requires that A < 1 for all k.
A2 = AA∗ = (1 − µ) + µe−ik∆x (1 − µ) + µeik∆x = (1 − µ)2 + µ(1 − µ)(eik∆x + e−ik∆x ) + µ2
= 1 − 2µ + 2µ(1 − µ) cos k∆x + 2µ
2 2 = 1 − 2µ(1 − cos k∆x) + 2µ (1 − cos k∆x)
k ∆x
(1 − µ)µ
= 1 − 4 sin2
2 (5.33)
(5.34)
(5.35)
(5.36)
(5.37) It is now clear that A2 ≤ 1 if µ(1 − µ) > 0, i.e. 0 ≤ µ ≤ 1. It is the same stability
criterion derived via the matrix analysis procedure. 5.6 Modiﬁed Equation The truncation error series used to establish the consistency of the scheme can be used
to extract additional information about the expected behavior of the numerical scheme. 78 CHAPTER 5. NUMERICAL SOLUTION OF PDE’S This is motivated by the observation that the ﬁnite diﬀerence scheme is in fact solving a
perturbed form of the original equation. Equations 5.9 and 5.12 establish that the FTBS
scheme approximates the advection equation to ﬁrst order, O(∆t, ∆x) to the advection
equation. They also show that FTBS approximates the following equation to second
order in time and space:
[vt + cvx ] = − c∆x
∆t
vtt +
vxx + O(∆t2 , ∆x2 ).
2
2 (5.38) The second term on the right hand side of equation 5.38 has the form of a diﬀusion
like operator, and hence we expect it to lead to a gradual decrease in the amplitude
of the solution. The interpretation of the time derivative term is not simple. The way
to proceed is to derive an expression for vtt in terms of the spatial derivative. This is
achieved by diﬀerentiating equation 5.38 once with respect to time and once with respect
to space to obtain:
∆t
c∆x
vttt +
vtxx + O(∆t2 + ∆x2 )
2
2
c∆x
∆t
vxxx + O(∆t2 + ∆x2 )
= − vttx +
2
2 vtt + cvxt = −
vtx + cvxx (5.39)
(5.40) Multiplying equation 5.40 by −c and adding it to equation 5.39 we get:
vtt = c2 vxx c∆x
∆t
(−vttt + cvtxx ) +
(vxxt − cvxxx ) + O(∆t2 , ∆x2 )
2
2 (5.41) Inserting this ﬁrst order approximation to vt t back in equation 5.38 we obtain the following modiﬁed equation.
vt + cvx = c
(∆x − c∆t) vxx + O(∆x2 , ∆x∆t, ∆t2 )
2 (5.42) Equation 5.42 is more informative then its earlier version, equation 5.38. It tells us that
the leading error term in ∆t, ∆x behaves like a second order spatial derivative whose
coeﬃcient is given by the pseudo, or numerical, viscosity νn , where
νn = c
(∆x − c∆t) .
2 (5.43) If νn > 0 we expect the solution to be damped to leading order, the numerical scheme
behaves as a advectiondiﬀusion equation, one whose viscous coeﬃcient is purely an
artifact of the ﬁnite diﬀerence discretization. If the numerical viscosity is negative,
νn < 0, the solution will be ampliﬁed exponentially fast. The stability condition that
νn > 0 is nothing but the usual stability criterion we have encountered earlier, namely
that c > 0 and µ = c∆t/∆x < 1.
A more careful analysis of the truncation error that includes higher powers of ∆t, ∆x
yields the following form:
vt + cvx = c∆x2
c∆x
(1 − µ)vxx −
(2µ2 − 3µ +1)vxxx + O(∆t3 , ∆t2 ∆x, ∆t∆x2 , ∆x3 ) (5.44)
2
6 5.6. MODIFIED EQUATION 79 The third derivative term is indicative of the presence of dispersive errors in the numerical
solution; the magnitude of these errors is ampliﬁed by the coeﬃcient multiplying the third
order derivative. This coeﬃcient is always negative in the stability region 0 ≤ µ ≤ 1.
One can expect a lagging phase error with respect to the analytical solution. Notice also
that the coeﬃcients of the higher order derivative on the right hand side term go to zero
for µ = 1. This “ideal” value for the time step makes the scheme at least third order
accurate according to the modiﬁed equation; in fact it is easy to convince one self on the
basis of the characteristic analysis that the exact solution is recovered.
Notice that the derivation of the modiﬁed equation uses the Taylor series form of
the ﬁnite diﬀerence scheme, equation 5.9, rather then the original partial diﬀerential
equations to derive the estimates for the high order derivative. This is essential to
account for the discretization errors. The book by Tannehill et all 1997 discusses a
systematic procedure for deriving the higher order terms in the modiﬁed equation. 80 CHAPTER 5. NUMERICAL SOLUTION OF PDE’S Chapter 6 Numerical Solution of the
Advection Equation
6.1 Introduction We devote this chapter to the application of the notions discussed in the previous chapter
to investigate several ﬁnite diﬀerence schemes to solve the simple advection equation.
This equation was taken as an example to illustrate the abstract concepts that frame
most of the discussion on ﬁnite diﬀerence methods from a theoretical perspective. These
concepts we repeat are consistency, convergence and stability. We will investigate several
common schemes found in the literature, and we will investigate their amplitude and
phase errors more closely. 6.2 Donor Cell scheme The donor cell scheme is essentially the FTBS scheme seen earlier. The only distinguishing feature of the donorcell scheme is that it allows the switching of the spatial ﬁnite
diﬀerence according to the sign of the advecting velocity c. A compact way of writing
the scheme is:
un+1 − un c + c un − un−1 c − c un+1 − un
j
j
j
j
j
j
+
+
=0
∆t
2
∆x
2
∆x (6.1) For c > 0 the scheme simpliﬁes to a FTBS, and for c < 0 it becomes a FTFS (forward
time and forward space) scheme. Here we will consider solely the case c > 0 to simplify
things. Figure 6.1 shows plots of the ampliﬁcation factor for the donor cell scheme. Prior
to discussing the ﬁgures we would like to make the following remarks. 6.2.1 Remarks 1. The scheme is conditionally stable since the time step cannot be chosen independently of the spatial discretization and must satisfy ∆t ≤ ∆tmax = c/∆x.
81 82 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION
A for FTBS µ = 0.25, 0.5, and 0.75
1 0.9
µ=0.25,0.75 0.8 µ=0.50 0.7 A 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 1.4 µ=0.75
1.2 µ=0.5,1.0 1 Φ/Φ a 0.8 0.6 0.4 µ=0.25 0.2 0 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 Figure 6.1: Amplitude and phase diagram of the donor cell scheme as a function of the
wavenumber 6.2. DONOR CELL SCHEME 83 2. The wavelength appearing in the VonNeumann stability analysis has not been
speciﬁed yet. Small values of k correspond to very long wavelegths, i.e. Fourier
modes that are well represented on the computational grid. Large values of k correspond to very short wavelength. This correspondence is evident by the expression
k∆x = 2π ∆x/λ, where λ is the wavelength of the Fourier mode. For example, a
twenty kilometers wave represented on a grid with ∆x = 2 kilometers would have
10 points per wavelength and its k∆x = 2π 2/10 = 2π/5.
3. There is an lower limit on the value of the shortest wave representable on a discrete
grid. This wave has a wavelength equal to 2∆x and takes the form of a seesaw
function; its k∆x = π . Any wavelength shorter then this limit will be aliased
into a longer wavelegth. This phenomenon is similar to the one encountered in the
Fourier analysis of time series where the Nyquist limit sets a lower bound to the
smallest measurable wave period.
4. In the previous chapter we have focussed primarily on the magnitude of the ampliﬁcation factor as it is the one that impacts the issue of stability. However, additional
information is contained in the expression for the ampliﬁcation factor that relates to
the dispersive properties of the ﬁnite diﬀerence scheme. The analytical expression
for the ampliﬁcation factor for a Fourier mode is
Aa = e−ick∆t. (6.2) Thus the analytical solution expects a unit ampliﬁcation per time step, Aa  = 1,
and a change of phase of φa = −ck∆t = −µk∆x. The ampliﬁcation factor for the
donor cell scheme is however:
A = Aeiφ , k ∆x
,
2
µ sin k∆x
1 − µ(1 − cos k∆x) A = 1 − µ(1 − µ)4 sin2
φ = tan−1 (6.3)
(6.4)
(6.5) where φ is the argument of the complex number A. The ratio of φ/φa gives the
relative error in the phase. A ratio less then 1 means that the numerical phase
error is less then the analytical one, and the scheme is decelerating, while a ratio
greater then indicates an accelerating scheme. We will return to phase errors later
when we look at the dispersive properties of the scheme.
5. The donor cell scheme for c positive can be written in the form:
un+1 = (1 − µ)un + µun−1
j
j
j (6.6) which is a linear, convex (for 0 ≤ µ ≤ 1), combination of the two values at the
previous time levels upstream of the point (j, n). Since the two factors are positive
we have
min(un , un−1 ) ≤ un+1 ≤ max(un , un−1 ),
(6.7)
j
j
j
j
j 84 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION
In plain words the value at the next time level cannot exceed the maximum of the
two value upstream, nor be less then the minimum of these two values. This is
referred to as the monotonicity property. It plays an important role in devising
scheme which do not generate spurious oscillation because of underresolved gradients. We will return to this point several times when discussing dispersive errors
and special advection schemes. Figure 6.1 shows A and φ/φa for the donor cell scheme as a function of k∆x for
several values of the Courant number µ. The long waves (small k∆x are damped the
least for 0 ≤ µ ≤ 1 whereas high wave numbers k∆x → 0 are damped the most. The
most vigorous damping occurs for the shortest wavelength for µ = 1/2, where the donorcell scheme reduces to an average of the two upstream value, the ampliﬁcation factor
magnitude is then A = 0, i.e. 2∆x waves are eliminated after a single time step. The
ampliﬁcation curves are symmetric about µ = 1/2, and damping lessens as µ becomes
smaller for a ﬁxed wavelength. The dispersive errors are small for long waves; they are
decelerating for all wavelengths for µ < 1/2 and accelerating for 1/2 ≤ µ ≤ 1; they reach
their peak acceleration for µ = 3/4. 6.3 Backward time centered space (BTCS) In this scheme the terms in the equations are evaluated at time (n + 1). For a two time
level scheme this translates into a backward euler diﬀerence for the time derivative. We
use a centered diﬀerence in space to increase the order of the spatial approximation. This
leads to the equations:
+1
+1
un+1 − un−1
un+1 − un
j
j
j
j
+c
=0
(6.8)
∆t
2∆x 6.3.1 Remarks 1. truncation error The Taylor series analysis (expansion about time level n + 1)
leads to the following equation:
ut + cux = − − ∆t
c∆x2
∆ t2
utt +
uxxx +
uttt + O(∆t3 , ∆x4 )
2
3
6 (6.9) The leading truncation error term is O(∆t, ∆x2 ), and hence the scheme is ﬁrst
order in time and second order in space. Moreover, the truncation error goes to
zero for ∆t, ∆x → 0, and hence the scheme is consistent.
2. The Von Neumann stability analysis leads to the following ampliﬁcation factor:
A=
A =
φ
φa = 1 − iµ sin k∆x
1 + µ2 sin2 k∆x
1
1 + µ2 sin2 k∆x (6.10)
< 1, for all µ, k∆x tan−1 (−µ sin k∆x)
−µk∆x (6.11)
(6.12) 6.3. BACKWARD TIME CENTERED SPACE (BTCS) 85 A for BTCS
1
µ=0.25 µ=0.50 0.8 A 0.9 µ=0.75 0.7 µ=1.00 0.6 0.5
µ=2.00
0.4 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 1 0.9
0.25
0.8
0.50 0.7 0.6
Φ/Φ a 0.75 0.5
1.00 0.4 0.3
2.00
0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 Figure 6.2: Amplitude and phase diagrams of the BTCS scheme as a function of the
wavenumber 86 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION
The scheme is unconditionally stable since A < 1 irrespective of the time step
∆t. By the LaxRichtmeyer theorem the consistency and stability of the scheme
guarantee it is also convergent.
3. The modiﬁed equation for BTCS is
ut + cux = c2 ∆t
uxx −
2 c∆x2 c3 2
+ ∆t uxxx + . . .
6
6 (6.13) The numerical viscosity is hence always positive and lends the scheme its stable
and damping character. Notice that the damping increasing with increasing c and
∆ t.
4. Notice that the scheme cannot update the solution values a grid point at a time,
+1
since the values un+1 and un±1 are unknown and must be determined simultanej
j
ously. This is an example of an implicit scheme which requires the inversion of a
system of equation. Segregating the unknowns on the left hand side of the equation
we get:
µ +1
µ +1
(6.14)
− un−1 + un+1 + un+1 = un
j
j
j
2
2j
which consititutes a matrix equation for the vector of unknowns at the next time
level. The equation in matrix forms are: −µ
2 1
−µ
2 0 µ
2 1
−µ
2 µ
2 1 µ
2 u
1 .
. . uj −1 uj u j +1 . .
. uN n+1 n u1
.
.
. u j −1 = uj uj +1 .
.
. uN (6.15) The special structure of the matrix is that the only nonzero entries are those along
the diagonal, and on the ﬁrst upper and lower diagonals. This special structure is
referred to as a tridiagonal matrix. Its inversion is far cheaper then that of a full
matrix and can be done in O(N ) addition and multiplication through the Thomas
algorithm for tridiagonal matrices; in contrast, a full matrix would require O(N 3 )
operations. Finally, the ﬁrst and last rows of the matrix have to be modiﬁed to
take into account boundary conditions. We will return to the issue of boundary
conditions later.
Figures 6.2 shows the magnitude of the ampliﬁcation factor A for several Courant
numbers. The curves are symmetric about k∆x = π/2. The high and low wave
numbers are the least damped whereas the intermediate wave numbers are the
most damped. The departure of A from 1 deteriorates with increasing µ. Finally
the scheme is decelarating for all wavenumbers and Courant numbers, and the
deceleration deteriorates for the shorter wavelengths. 6.4. CENTERED TIME CENTERED SPACE (CTCS) 87 1.00 1 0.9
0.25
0.8 0.50
0.75 0.7 Φ/Φ a 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 Figure 6.3: Phase diagrams of the CTCS scheme as a function of the wavenumber 6.4 Centered time centered space (CTCS) A simple and popular explicit second order scheme in space and time is the centered
time and centered space scheme. This is a three time level scheme and takes the form:
n
n
uj +1 − uj −1
un+1 − un−1
j
j
+c
=0
2∆t
2∆x 6.4.1 (6.16) Remarks 1. truncation error The Taylor series analysis leads to the following equation:
ut + cux = − − c∆x2
∆ t2
uttt +
uxxx + O(∆t4 , ∆x4 )
3
3 (6.17) The leading truncation error term is O(∆t2 , ∆x2 ), and hence the scheme is second
order in time and space. Moreover, the truncation error goes to zero for ∆t, ∆x →
0, and hence the scheme is consistent. 88 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION
2. The Von Neumann stability analysis leads to a quadratic equation for the ampliﬁcation factor: A2 + 2µ sin k∆xA − 1 = 0. Its two solutions are:
A± = −iµ sin k∆x ± 1 − µ2 sin2 k∆x A±  = 1, for all µ < 1
φ
−µ sin k∆x
1
tan−1
=
φa
−µk∆x
1 − µ2 sin2 k∆x (6.18)
(6.19)
(6.20) The scheme is conditionally stable for µ < 1, and its ampliﬁcation is neutral
since A = 1 within the stability region. An attribute of the CTCS scheme is
that its ampliﬁcation factor mirror the neutral ampliﬁcation of the analytical solution. By the LaxRichtmeyer theorem the consistency and stability of the scheme
guarantee it is also convergent.
3. The modiﬁed equation for CTCS is
ut + cux = c∆x2 2
c∆x4
(µ − 1)uxxx −
(9µ4 − 10µ2 + 1)uxxxxx + . . .
6
120 (6.21) The even derivative are absent from the modiﬁed equation indicating the total
absence of numerical dissipation. The only errors are dispersive in nature due to
the presence of odd derivative in the modiﬁed equation.
4. The model requires a starting procedure to kick start the computations. It also
has a computational mode that must be damped.
5. Figures 6.3 shows the phase errors for CTCS for several Courant numbers. All
wave numbers are decelerating and the shortest wave are decelerated more then
the long waves. 6.5 Lax Wendroﬀ scheme The idea behind the LaxWendroﬀ scheme is to keep the simplicity of two time level
schemes while attempting to increase the order of accuracy in space and time to second
order. This is possible if derivatives in time are translated to derivatives in space. The
Taylor series in time about time level n is:
un+1 = un + ∆tut +
j
j ∆ t3
∆ t2
utt +
uttt
2
6 (6.22) From the PDE we know that ut = −cux . What we need to complete the second order accuracy in time is a second order expression for utt . This can be obtained by diﬀerentiating
the advection equation with respect to time to yield:
utt = −cuxt = −c(ut )x = −c(−cux )x = c2 uxx (6.23) The Taylor series in time takes the form:
un+1 = un − c∆tux +
j
j ∆ t3
c2 ∆t2
uxx +
uttt
2
6 (6.24) 6.5. LAX WENDROFF SCHEME 89
A for Lax−Wendroff µ=1.00 1 0.9 µ=0.25 0.8 0.7 A 0.6 µ=0.50 0.5 0.4 0.3 0.2 0.1 µ=0.75
0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 1.4 1.2 1.00 1
0.75 0.25
0.50 Φ/Φ a 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5
k ∆ x/π 0.6 0.7 0.8 0.9 1 Figure 6.4: Amplitude and phase diagrams of the LaxWendroﬀ scheme as a function of
the wavenumber 90 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION All that remains to be done is to use high order approximations for the spatial derivatives
ux and ux x. We use centered derivatives for both terms as they are second order accurate
to get the ﬁnal expression:
un+1 − un
un+1 − un−1 c2 ∆t un+1 − 2un + un−1
j
j
j
j
j
j
j
= −c
+
2
∆t
2∆x
2
∆x 6.5.1 (6.25) Remarks 1. truncation error The Taylor series analysis (expansion about time level n + 1
leads to the following equation:
ut + cux = − − ∆ t2
c∆x2
uttt +
uxxx + O(∆t4 , ∆x4 )
3
3 (6.26) The leading truncation error term is O(∆t2 , ∆x2 ), and hence the scheme is second
order in time and space. Moreover, the truncation error goes to zero for ∆t, ∆x →
0, and hence the scheme is consistent.
2. The Von Neumann stability analysis leads to:
A = 1 − µ2 (1 − cos k∆x) − iµ sin k∆x
2 A
φ
φa 2 2 2 2 = [1 − µ (1 − cos k∆x)] + µ sin k∆x,
−µ sin k∆x
1
=
tan−1
−µk∆x
1 − µ2 (1 − cos k∆x) (6.27)
(6.28)
(6.29) The scheme is conditionally stable for µ < 1. By the LaxRichtmeyer theorem
the consistency and stability of the scheme guarantee it is also convergent.
3. The modiﬁed equation for Lax Wendroﬀ is
ut + cux = c∆x3
c∆x2 2
(µ − 1)uxxx −
µ(1 − µ2 )uxxxx + . . .
6
8 (6.30) 4. Figures 6.4 shows the amplitude and phase errors for the Lax Wendroﬀ schemes.
The phase errors are predominantly lagging, the only accelerating errors are those
of the short wave at relatively high values of the Courant number. 6.6 Numerical Dispersion Consistency and stability are the ﬁrst issues to consider when contemplating the solution
of partial diﬀerential equations. They address the theoretical questions of convergence
in the limit of improving resolution. They should not be the last measure of performance, however, as other error measures can be of equal importance. For hyperbolic
equations, where wave dynamics are important, it is critical to consider the distortion
of wave propagation by the numerical scheme. Although, we have looked at the phase
characteristic of the scheme derived so far, it was hard to get an intuitive feel for the
impact on the wave propagation characteristics. The aim of this section is to address the
issue of the numerical dispersion relation, and to compare it to the dispersion relation of
the continuous equations. We start with the latter. 6.6. NUMERICAL DISPERSION 6.6.1 91 Analytical Dispersion Relation The analytical dispersion relation for the wave equation can be obtained by looking for
periodic solutions in space and time of the form uei(kx−ωt) where k is the wavenumber
˜
and ω the corresponding frequency. Inserting this expression in equation 5.3 we get the
dispersion relation:
ω = ck
(6.31)
The associate phase speed, Cp , and group velocity, Cg , of the system is as follows
Cp =
Cg = ω
=c
k
∂ω
=c
∂k (6.32)
(6.33) The two velocities are constant and the system is nondispersive, i.e. all waves travels
with the same phase speed regardless of wavenumber. The group velocity is also constant
in the present case and reﬂects the speed of energy propagation. One can anticipate that
this property will be violated in the numerical discretization process based on what we
know of the phase error plots; there it was shown that phase errors are diﬀerent for the
diﬀerent wave number. We will make this assertion clearer by looking at the numerical
dispersion relation. 6.6.2 Numerical Dispersion Relation: Spatial Diﬀerencing To keep the algebra tractable, we assume that only the spatial dimension is discretized
and the time dimenion is kept continuous. The semidiscrete form of the following
schemes are:
1. Centered second order scheme
uj +1 − uj −1
=0
2∆x (6.34) 8(uj +1 − uj −1 ) − (uj +2 − uj −2 )
=0
12∆x (6.35) ut +
2. Centered fourth order scheme
ut + 3. Centered sixth order scheme
ut + 45(uj +1 − uj −1 ) − 9(uj +2 − uj −2 ) + (uj +3 − uj −3 )
=0
60∆x 4. Donor cell (6.36) uj − uj − 1
=0
∆x (6.37) 2uj +1 + 3uj − 6uj −1 ) + uj −2
=0
6∆x (6.38) ut +
5. Third order upwind
ut + 92 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION The dispersion for these numerical scheme can be derived also on the basis of periodic
solution of the form uj = uei(kxj −σ) . The biggest diﬀerence is of course that the Fourier
˜
expansion is discrete in space. The following expression for the phase velocity can be
derived for the diﬀerent schemes:
CD2
CD4
CD6
Donor
Third Upwind σ
k
σ
k
σ
k
σ
k
σ
k =c
=c
=c
=c
=c sin k∆x
k ∆x
8 sin k∆x − sin 2k∆x
6k∆x
sin 3k∆x − 9 sin 2k∆x + 45 sin k∆x
30k∆x
sin k∆x − i(1 − cos k∆x)
k ∆x
(8 sin k∆x − sin 2k∆x) − i2(1 − cos k∆x)2
k ∆x (6.39) Several things stand out in the numerical dispersion of the various schemes. First, all
of them are dispersive, and hence one expects that a wave form made up of the sum of
individual Fourier components will evolve such that the fast travelling wave will pass the
slower moving ones. Second, all the centered diﬀerence scheme show a real frequency,
i.e. they introduce no amplitude errors. The oﬀcentered schemes on the other hand
have real and imaginary parts. The former inﬂuences the phase speed whereas the
former inﬂuences the amplitude. The amplitude decays if I m(σ ) < 0, and increases for
I m(σ ) > 0. Furthermore, the upwind biased schemes have the same real part as the next
higher order centered scheme; thus their dispersive properties are as good as the higher
order centered scheme except for the damping associated with their imaginary part (this
is not necessarily a bad things at least for the short waves).
Figure 6.5 shows the dispersion curve for the various scheme discussed in this section
versus the analytical dispersion curve (the solid straight line). One can immediately see
the impact of higher order spatial diﬀerencing in improving the propagation characteristics of the intermediate wavenumber range. As the order is increased, the dispersion
curves rise further towards the analytical curve, particularly near k∆x > π/2, hence a
larger portion of the spectrum is propagating correctly. The lower panel shows the impact
of biasing the diﬀerencing towards the upstream side. The net eﬀect is the introduction
of numerical dissipation. The latter is strongest for the short waves, and decreases with
the order of the scheme.
Figure 6.6 shows the phase speed (upper panel) and group velocity (lower panel) of
the various schemes. Again it is evident that a larger portion of the wave spectrum is
propagating correctly whereas as the order is increased. None of the schemes allows the
shortest wave to propagate. The same trend can be seen for the group velocity plot.
However, the impact of the numerical error is more dramatic there since the short waves
have negative group velocities, i.e. they are propagating in the opposite direction. This
trend worsens as the accuracy is increased. 6.6. NUMERICAL DISPERSION 93 Numerical Dispersion of CD of 2, 4 and 6 order
3.5
3
2.5 ω 2
FE1 1.5 CD6
CD4 1 CD2
0.5
0
0 0.2 0.4 k /(∆ xπ) 0.6 0.8 1 Imaginary part of frequency for Upstream Difference of 1, and 3 order
2
1.8
1.6
1.4 ωi 1.2
1 1 0.8
0.6
0.4
3
0.2
0
0 0.2 0.4 k /(∆ xπ) 0.6 0.8 1 Figure 6.5: Dispsersion relation for various semidiscrete schemes. The upper panel
shows the real part of the frequency whereas the bottom panel shows the imaginary part
for the ﬁrst and third order upwind schemes. The real part of the frequency for these
two schemes is identical to that of the second and fourth order centered schemes. 94 CHAPTER 6. NUMERICAL SOLUTION OF THE ADVECTION EQUATION Numerical phase speed of various spatial discretization
1
0.9
0.8
FE1 0.6 CD6 0.5 CD4 cnum/can 0.7 0.4
CD2 0.3
0.2
0.1
0
0 0.2 0.4 k ∆ x/π 0.6 0.8 1 Group velocity
1
0.5
0 g −0.5
CD2 −1.5 CD4 −2 c −1 CD6 −2.5
−3
0 FE1
0.2 0.4 k∆ x/π 0.6 0.8 1 Figure 6.6: Phase speed (upper panel) and Group velocity (lower panel) for various
semidiscrete schemes. Chapter 7 Finite Volume Method
. This chapter focusses on introducing ﬁnite volume method for the solution of partial
diﬀerential equations. These methods are in widespread use for their robustness, their
intuitive formulation, and oﬀer some clear advantages, primarily ensuring the conservation of a quantity of interest. We will take look at the formulation, discretization, and
coding of these methods. 7.1 The partial diﬀerential equation The partial diﬀerential equation we will focus on is a scalar equation that represents the
transport of a substance under the inﬂuence of advection by the air ﬂow and mixing.
The transport equation is frequently written in the advective form:
∂T
+ u · ∇T = ∇ · (α∇T )
∂t (7.1) where T is the subtance transported, e.g. temperature, humidity or a pollutant concentration, u is the velocity ﬁeld presumed known, and α is the diﬀusion coeﬃcient and
which can represent either molecular diﬀusion or eddy mixing. The advective form can
be interpreted as the time evolution of the T ﬁeld along characteristic lines given by
dx = u (and in the absence of diﬀusive eﬀects or source terms, T is constant). The
dt
advective form is thus closest to a Lagrangian description of the ﬂow where one follows
individual particles.
A slightly diﬀerent form of the equation called the conservative form can be derived
and forms the starting point for the derivation of ﬁnite volume methods. The connection
between the advective and conservation is mediated by a statement of conservation of
mass. Indeed, the velocity ﬁeld cannot be arbitrary and must satisfy some sort of mass
conservation equation. Here we will assume the ﬂow to be incompressible so that its
mass conservation equation reduces to:
∇·u=0 (7.2) Multiplying the continuity equation by T , adding it to the resultant equations to the
advective form, and recalling that u ·∇T + T ∇· u = ∇· (uT ) we can derive the conservative
95 96 CHAPTER 7. FINITE VOLUME METHOD form of the transportdiﬀusion equation:
∂T
+ ∇ · (uT ) = ∇ · (α∇T ).
∂t (7.3) The advection and conservation forms are both valid statements at every point in
the domain. Although the two forms are equivalent in the continuum, they express
diﬀerent aspect of physical laws. The advection form describes the evolution of T along
ﬂuid trajectories whereas the conservation form describe the conservation of T at every
point in the domain (see section 7.2 for an explanation of this point of view). It is
also important to remember that the two statements hold simultaneously thanks to the
conservation of mass.
Although the advection and conservation equation are equivalent in the continuum
case, the equivalency maybe broken in the discretization process. Hence discretizing
the advective or conservation form will lead to the approximate enforcement of slightly
diﬀerent physical laws. In some application, the issue of conservation is essential, so
that the T is conserved for long simulation times. This concern stems not only from
physical considerations but also for the need to account for the sources and sinks of T
in long calculations, or in complex simulations. Examples include climate simulations:
their simulation time is centuries, and it is important to account for all the sinks of heat
or carbon dioxide. Another example are simulation of chemical reactions or combustion.
Hence, in situation where conservation is paramount it is natural to discretize the continuous form starting from the conservation statement, and ensuring that the discretization
does not introduce spurious sources.
Finite volume methods are ideally suited to enforce conservation laws in the discrete
case. They have a further virtue: for solution that develop discontinuities and where the
spatial derivative may fail to exist at a number of location, the ﬁnite volume procedure
remains valid as it acts on an integrated form of the equations as we will see shortly. 7.2 Integral Form of Conservation Law The partial diﬀerential equation 7.3 is valid at all points in the domain which we could
consider as inﬁtesimal volumes. Anticipating that inﬁnitesimal discrete volumes are
unaﬀordable and would have to be ”inﬂated” to a ﬁnite size, we proceed to derive the
conservative form for a ﬁnite volume δV bounded by a surface δS as shown in ﬁgure 7.1.
Integrating equation 7.3 over the control volume δV we get:
δV d
dt ∂T
dV +
∂t
T dV
δV δV ∇ · (uT ) dV = n · uT dV = +
δS ∇ · (α∇T ) dV (7.4) δV n · α∇T dV (7.5) δS Remarks
• We assume the volume δV to be ﬁxed in space so we can interchange the order
of integration in space and diﬀerentiation in time. The interpretation of the ﬁrst
integral on the left hand side of equation 7.5 is now simple: it is the time rate of
change of the T budget inside volume δV . 7.2. INTEGRAL FORM OF CONSERVATION LAW 97 n Tu '$ δV δS &% Figure 7.1: Sketch of the volume δV and its bounding surface δS .
• We have used the Gaussdivergence theorem to change the volume integrals of the
ﬂux and diﬀusion divergence into surface integrals. Here n is the outward unit
normal to the surface δS . The surface integral on the left hand side accounts for
the advective ﬂux carrying T in and out of the volume δV across the surface δS ;
the one on the right hand side accounts for the diﬀusive transport of T across δS .
• Equation 7.5 lends itself to simple physical interpretation: the rate of change of
the T budget in δV is equal to the rate of transport of T through δS by advective
ﬂuxes (transport by the ﬂow, wind, current) and diﬀusive ﬂuxes.
• If the volume δV is closed to advection or diﬀusion: u.n = 0 and ∇T · n = 0, then
the rate of change is zero and the budget of T is conserved within δV .
• The volume δV is so far arbitrary and we have not assigned it a speciﬁc shape or
size. Actually equation 7.5 applies to any volume δV whether it is a computational
cell, an entire ocean basin, or Earth’s atmosphere.
• If there are additional physical processes aﬀecting the budget of T , such as sources
or sinks within δV these should be accounted for also. No additional process is
considered here.
• The derivatives appearing in the diﬀential equation 7.1 are: ﬁrst order for the
advective term, and second order for the diﬀusion term, whereas the equivalent
terms in the integral form have only a zeroth order derivative and ﬁrst order
derivative respectively. This lowering of the derivative order is important in dealing
with solution which change so rapidly in space that the spatial derivative does not
exist. Examples include supersonic shock waves in the atmosphere or hydraulic
jump in water where ﬂuid properties such as density or temperature change so fast
that it appears discontinuous. Discontinuous function do not have derivatives at
the location of discontinuity and mathematically speaking the partial diﬀerential
form of the conservation equation is invalid there even though the conservation
law underlying it is still valid. Special treatment is generally required for treating
discontinuities and reducing order of the spatial derivative helps simplify the special
treatment. For these reasons Finite volumes are preferred over ﬁnite diﬀerences to
solve problems whose solution exhibit local discontinuities.
A slightly diﬀerent form of equation 7.5 can be derived by introducing the average of T
in δV and which we refer to as T and is deﬁned as:
T= 1
δV T dV.
δV (7.6) 98 CHAPTER 7. FINITE VOLUME METHOD The integral conservation law can now be recast as a time evolution equation for T :
δV 7.3 dT
+
dt δS n · uT dV = δS n · α∇T dV (7.7) Sketch of Finite Volume Methods Equation 7.5 and 7.7 are exact and no approximation was necessary in their derivation.
In a numerical model, the approximation will be introduced by the temporal integration
of the equations, and the need to calculate the ﬂuxes in space and time. The traditional
ﬁnite volume method takes equation 7.7 as its starting point. The domain is ﬁrst divided
into computational cells δVj where the cell average of the function is known. The
advection and diﬀusion ﬂuxes are calculated in two steps:
• Function reconstruction The advective ﬂuxes require the calculation of the function values at cell edges, while the diﬀusive ﬂuxes require the calculation of the
function derivative at cell edges. The latter are obtained from approximating the
function T with a polynomial whose coeﬃcients are determined by the need to
recover the cell averages over a number of cells.
P ˜
T= an φ(x) (7.8) n=1 ˜
T dV
δVj +m = δVj +m T j +m , m = 0, 1, 2, . . . , P − 1 (7.9) where δVj +m are P cells surrounding the cell δVj and where the φn are P suitably chosen interpolation functions (generally simple polynomials). The unknown
coeﬃcients an are recovered simply from the simultaneous algebraic equation
P Am,n an = δVj +m T j +m , Am,n = n=1 δVj +m φn dV. (7.10) Once the coeﬃcients an are known it becomes possible to evaluate the function
on cell edges with a view to compute the ﬂux integrals. The error in the spatial
approximation is primarily due to equation 7.8.
• Evaluation of the integrals. The integrals are usually evaluated numerically
using Gauss type quadrature. The approximation function φn generally determine
the number of quadrature points so that the quadrature is exact for polynomials of
degree P . No error is then incurred during the spatial integration. One can then
write:
Q Fj =
q =1 ˜
u · nT δSj  xq ωk (7.11) where ωk are the weights of the Gaussian quadrature (as appropriate for multidimensional integration, xq are the Q quadrature points, and δSj are the mapping
factors of the surface δSj 7.4. FINITE VOLUME IN 1D 99 • Temporal integration The ﬁnal source of error originates from the temporal
integration whereby the ﬂuxes are used to advance the solution in time using a
time marching procedure a la Forward Euler, one of the Runge Kutta methods,
or the AdamsBashforth class of methods. The time integration cannot be chosen
independtly of the spatial approximation, but is usually constrained by stability
considerations. 7.4 Finite Volume in 1D
r 1
1
2 r r 2
3
2 r 3
5
2 ···
7
2 r E j−1 j− 3
2 r E j j− 1
2 cell r j+1
j+ 1
j
2 + 3
2 r r M −1 r M j Figure 7.2: Discretization of a onedimensional domain into computational cells of width
δx. The cell centers are indicated by ﬁlled circles and cell edges by vertical lines ; the
cell number is shown above the cells while the numbers below the line indicate cell edges.
To illusrate the application of the procedure outlined in the previous section it is
best to consider the simple case of onespace dimension. In that case the cells are line
segments as shown in ﬁgure 7.2. The cell volumes reduces to the width of the segment δx
which we assume to be the same for all cells in the following. The ﬂux integrals reduce
to evaluation of the term at the cell edges. The conservation law can now be written as:
Fj + 1 − Fj − 1
Dj + 1 − Dj − 1
dT j
2
2
2
2
=−
+
dt
δx
δx (7.12) where Fj + 1 = uT and Dj + 1 = αTx are the advective ﬂux and diﬀusive ﬂuxes at xj + 1 .
2 7.4.1 2 2 Function Reconstruction The function reconstruction procedures can be developed as follows prior to timeintegration.
Assume
P
x − xj
T=
an ξ n , where ξ =
(7.13)
δxj
n=0
Here an are the P +1 coeﬃcients that need to be determined from P +1 conditions on the
averages of the polynomials over multiple cells. The origin and scale of the coordinate
system have been shifted to the centre of cell j and scaled by its width for convenience so
that the left and right edges are located at a scaled length of 1/2 and 1/2, respectively.
The entries of the matrix are now easy to ﬁll since
Am,n = xm+ 1
2 xm− 1
2 = δxj
n x − xj − 1
2 n−1 δxj xm+ 1 − xj − 1
2 δxj 2 dx, for n = 1, 2, . . . , P + 1 n − xm− 1 − xj − 1
2 δxj 2 (7.14) n (7.15) 100 CHAPTER 7. FINITE VOLUME METHOD = δxj 7.4.2 n
n
ξm+ 1 − ξm− 1
2 2 , with ξm± 1 = n xm± 1 − xj
2 (7.16) δxj 2 Piecewise constant
r r r r r r j−1 j r j+1 r r M −1 r M Figure 7.3: Piecewise constant approximation.
The simplest case to consider is one where the function is constant over a cell as
shown in ﬁgure 7.3. Then we have the approximation:
T = a0 (7.17) Since there is only one unknown coeﬃcient, a0 , we can only impose a single constraint,
namely that the integral of T over cell j yields δxj T j :
xj + 1
2 xj − 1 a0 dx = δxj T j (7.18) 2 The solution is then simply a0 = T j . The function reconstruction at xj + 1 can be done.
2
There are a couple of things to notice ﬁrst with respect to implementing the solution
algorithm:
• The edge point two approximations are possible, one from the left cell j and one
from the right cell j + 1. For the purpose of deﬁning the advective ﬂux, a physically
intuitive justiﬁcation is that the information reaching the edge should come from
the upstream cell, i.e. the cell where the wind is blowing from. Thus we have:
if uj + 1 ≥ 0
2
if uj + 1 < 0 Tj
T j +1 Tj + 1 =
2 (7.19) 2 • Piecewise constant function have a zero spatial derivative, and hence piecewise
constants are useless for computing the function derivation for the diﬀusion term. 7.4.3 Piecewise Linear
r r r r r j−1 $$
$$$
$$$
r j r j+1 r Figure 7.4: Piecewise Linear Approximation. r M −1 r M 7.4. FINITE VOLUME IN 1D 101 An improvement over the constant approximation is to assume linear variations for
the function over two cells as shown in ﬁgure 7.4. Then we have the approximation:
T = a0 + a1 ξ (7.20) Since there are two unknown coeﬃcient, a0 and a1 , we need to impose two integral
constraints. The choice of reconstruction stencil is at our disposal; here we choose a
symmetric stencil consisting of cells j and j + 1 so that xj − 1 ≤ x ≤ xj + 3 ; equivalently
2 1
− 2 ≤ ξ ≤ 3 . The two constraints become:
2
1
2
1
−2
3
2 1
2 2 a0 + a1 ξ dξ = T j
a0 + a1 ξ dξ = (7.21) δxj +1
T j +1
δxj (7.22) Performing the integration we obtain the following system of equations for the unknowns:
10
11 a0
a1 Tj = δxj +1
T j +1
δxj (7.23) When the grid cells are of constant size the solution is simply:
a0 = T j , a1 = T j +1 − T j −1 (7.24) The linear interpolation is then
T = T j + T j +1 − T j ξ
= (1 − ξ )T j + ξ T j +1 , for − (7.25)
3
1
≤ξ≤
2
2 (7.26) 1
To evaluate the function at the cell edge ξj ± 1 it is enough to set ξ = ± 2 .
2 7.4.4 Piecewise parabolic For parabolic interpolation centered on the cells j − 1, j and j + 1, the polynomial takes
the form
T j +1 − T j −1
T j +1 − 2T j + T j −1 2
−T j +1 + 26T j − T j −1
+
ξ+
ξ
24
2
2
3
x − xj
3
(7.27)
for − ≤ ξ ≤ , ξ =
2
2
δx T= At xj + 1 the cell edge value can now be computed by setting ξ = 1/2 in the expression
2
above to get:
−T j −1 + 5T j + 2T j +1
Tj + 1 =
(7.28)
2
6 102 CHAPTER 7. FINITE VOLUME METHOD 7.4.5 Reconstruction Validation The validation of the characteristics of the diﬀerent reconstruction procedures will be
illustrated here by looking at some important examples. In order to characterize the
procedures accurately, we need to solve a problem with a known solution, and compare
the results to what the numerical scheme yields. The result is of course the error in the
reconstruction. We will look at several examples with distinct characteristics: function
with smooth variations, and functions with local kinks in the graphs. The kinks are
symptomatic of a break in the smoothness of the function, as in when the slope of
the graph changes abruptly (slope discontinuity) or when the value of the function itself
changes abruptly (function discontinuity). In all cases we will be interested in monitoring
the decrease of the error with increasing number of cells (decreasing δx) for diﬀerent
functions, and for each of the reconstruction procedures.
Inﬁnitely smooth proﬁle
Our ﬁrst example consists of reconstructing the function T = cos πx on the interval
−1 ≤ x ≤ 1, starting from its analytical cell average over cell j :
Tj = 1
δx xj + 1 2 xj − 1 cos πx dx = sin πxj + 1 − sin πxj − 1
2 2 πδx (7.29) 2 The result of the reconstruction using the piecewise constant, linear and parabolic reconstruction are shown by the symbols in ﬁgures 7.57.7, respectively, and the error at each
cell edge is given by the height diﬀerence between the solid curve (reference solution)
and the symbol.
Remarks
• The 4 cell discretization is obviously too coarse to represent the cosine wave properly, particularly for the piecewise constant case.
• There is a dramatic improvement in going from to linear to parabolic in the case of
the 8 cell discretication (top right panel of ﬁgures); whereas the piecewise constant
reconstruction still incurs substantial errors.
• The piecewise linear and parabolic reconstruction produce out of bound values
(i.e. outside the range of the initial data whose absolute value is bounded by 1) at
x = ±1 because extrapolation had to be used near the edges. This is a recurrent
theme in methods whose stencil exceed a single cell.
• The error decreases with increasing cell numbers (decreasing δx) for all three reconstruciton methods considered.
• It is visually apparent that the decrease in error is fastest for the piecewise parabolic
reconstruction then for the piecewise constant.
An important issue concerns the rate at which the error decreases for each method.
Figure 7.8 shows the convergence curves for the diﬀerent methods as the number of cells 7.4. FINITE VOLUME IN 1D 103 1 1 0.5 0.5 0 0 −0.5 −0.5 −1
−1 −0.5 0 0.5 1 −1
−1 1 1 −0.5 0 0.5 1 0 −0.5 0.5 0.5 0 0 1 0.5 −0.5 −0.5 −1
−1 −0.5 0 0.5 1 −1
−1 Figure 7.5: Reconstruction at cell edges using piecewise constant reconstruction with 4
(top left), 8 (top right), 16 (bottom left) and 32 (bottom right) cells. The solid curved
line is the exact solution, whereas the staircases represent the cells and the cell averages.
In the present case we have assumed that the wind is blowing from left to right except at
the right most cell. The * symbols are the values of T calculated by the reconstruction
procedure. The error is thus the height diﬀerence between the solid curve and the
symbols. 104 CHAPTER 7. FINITE VOLUME METHOD 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 1 −0.5 0 0.5 1 −0.5 −1 1 0 −0.5 0.5 0.5 0 0 1 0.5 −0.5 −1 −1 −0.5 0 0.5 1 −1 Figure 7.6: Same as ﬁgure 7.5 but using piecewise reconstruction interpolation across 2
cells. Onesided reconstruction was used at both endedges. 7.4. FINITE VOLUME IN 1D 105 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 1 −0.5 0 0.5 1 −0.5 −1 1 0 −0.5 0.5 0.5 0 0 1 0.5 −0.5 −1 −1 −0.5 0 0.5 1 −1 Figure 7.7: Same as ﬁgure 7.5 but using piecewise parabolic reconstruction across 3 cells.
One sided reconstruction was used for endedges and for the ﬁrst internal left edge. 106 CHAPTER 7. FINITE VOLUME METHOD
0 10 −1 10 −2 maxT−Te 10 −3 10 −4 10 −5 10 −6 10 0 1 10 2 3 10
10
number of cells=2/δ x 10 Figure 7.8: Convergence curves showing ﬁrst, second, and third order convergence for
the piecewise constant (black), linear (red) and parabolic (blue) reconstruction procedures, respectively. Reference slopes of N −1 , N −2 and N −3 are shown in dashed line for
comparison
.
1 1 1 0.5 0.5 0.5 0 0 0 −1 0 1 −1 0 1 −1 0 1 Figure 7.9: Quadratic reconstruction for a function with discontinuity using, from left to
right, 10, 20 and 40 cells, respectively, with maximum errors of 0.25, 0.1623, and 0.1580.
is increased. The slope of the curves on this loglog plot shows the order of the method.
For small enough δx, the slope for the constant, linear and parabolic reconstruction
asymptotes to 1, 2, and 3 respectively, indicating a convergence of O(δx), O(δx2 ) and
O(δx3 ).
Reconstructing Functions with Discontinuities
The function
T (x) = 1
x−a
x+a
− tanh
tanh
2
δl
δl (7.30) exhibit two transition zones at x = ±a of width δl as shown in ﬁgure 7.4.5. as δl becomes
smaller then the grid spacing, the transition zones appear like discontinuities in the function. High order reconstruction of functions with discontinuities is problematic because
of spurious oscillations as shown in ﬁgure 7.4.5 near the discontinuities. The maximum 7.5. FINITE VOLUME METHOD FOR SCALAR ADVECTION IN 2D 107 error norm decreases marginally as the number of cell is increased, and does not exhibit
the cubic decrease expected for smooth functions. Notice also that the quadratic reconstruction has produced function values outside the range of the original data (larger
then 1 at the left discontinuity and a negative value near the right discontinuity). These
oscillations are commonly known as Gibbs oscillations and their amplitude does not decrease with increasing resolution. These oscillations are a direct consequence of applying
the reconstruction across a stencil that includes a discontinuity. Notice that the ﬁrst
and second order reconstruction cannot produce out of range values, and as such are
preferable near sharp transition zones.
Many remedies have been proposed to mitigate the generation of these oscillations.
Their common threads is to: ﬁrst, test for the smoothness of the solution locally, and
second, vary the order of the reconstruction and/or the stencil to avoid generating these
discontinuities. In sharp transition zone loworder nonoscillatory schemes would be
used whereas in smooth regions, highorder reconstruction would be used for improved
accuracy. Here we explore one such procedure dubbed the Weighed Essentially NonOscillatory scheme (WENO), proposed by Shu and colleagues. 7.5 Finite Volume Method for Scalar Advection in 2D E k+1
k
k−1 E
E
E
E
E T
u
T
u
T
u
T
u
T
u
T
u
T E
E
E
E
E
E T
u
T
u
T
u
T
u
T
u
T
u
T j−1 E
E
E
E
E
E T
u
T
u
T
u
T
u
T
u
T
u
T j E
E
E
E
E
E T
u
T
u
T
u
T
u
T
u
T
u
T E
E
E
E
E
E T
u
T
u
T
u
T
u
T
u
T
u E
E
E
E
E
E T j+1 Figure 7.10: Cartesian ﬁnite volume grid with rectangular cells. The solid point represents the average of T over a cell while the arrows denote the x − y advective ﬂuxes into
a cell through its 4 edges.
The spatial discretization proceeds by dividing the domain into ﬁnite volume cells
whose shape is left to the user. Finite volume codes have commonly used triangular cells
(eg FVCOM) or rectangular cells. In the former the grid can be made unstructured and 108 CHAPTER 7. FINITE VOLUME METHOD resembling that used in ﬁnite element methods. In the rectangular case, the grids are
structured and look like ﬁnite diﬀerence grids. In recent year the atmospheric community
has spearheaded the development of hexagonal ﬁnite volume cells because they can be
used eﬃciently to discretize spherical surfaces. Here we focus exclusively on rectangular
cells as depicted in ﬁgure 7.10.
We are now concerned with assigning the diﬀerent terms appearing in equation 7.7.
Here we presume that diﬀusion is nonexistent (α = 0). We have the following remarks:
• The cell volume is actually an area in 2D and we denote by δA = ∆x∆y the area
of the cell where ∆x × ∆y are the cell sizes in the x × y directions.
• The cells can be referenced by a pair of indices (j, k) along the (x, y ) directions. The
cell center has coordinate (xj , yk ) and the cell walls are located at (xj ± ∆x , yk ± ∆y ).
2
2
• Each cell has four edges with constant normal along each. The outward unit
normal to cell (j, k) at x = xj + ∆x/2 points in the positive xdirection, and we
have u · n = u; whereas at x = xj − ∆x/2 it points in the negative xdirection and
we have u · n = −u. Similarly along y = yk + ∆y/2 we have u · n = v , whereas
along y = yk − ∆y/2 we have u · n = −v .
With these remarks we can now write down the ﬁnite volume equation for cell (j, k):
dT j,k
δA
dt +
+
=0 yk + ∆ y
2
yk − ∆ y
2 u xj + 1 , y T xj + 1 , y − u xj − 1 , y T xj − 1 , y dy v x, y k + 1 T x, y k + 1 − v x, y k − 1 T x, y k − 1 dx 2 x j + ∆x
2 x j − ∆j
2 2 2 2 2 2 2 2 (7.31) where xj ± 1 = xj ± ∆x , and yk± 1 = yk ± ∆y . In this form we see that the equation is
2
2
2
2
nothing but an accounting of the ﬂux entering/leaving cell j . For ease of notation we
denote the x, y ﬂuxes by (F, G) = (uT, vT ), and the equation can be rewritten as:
dT j,k
δA
dt +
+ yk + ∆ y
2
yk − ∆ y
2 x j + ∆x
2 x j − ∆j
2 F xj + 1 , y − F xj − 1 , y dy G x, y k + 1 − G x, y k − 1 dx = 0 2 2 2 2 An added wrinkle to the 2D ﬁnite volume formulation is the need to evaluate boundary integrals (which were not encountered in the 1D case). The silver lining is that all
the integrals have the form:
zm + ∆z
2 zm − ∆z
2 H (z ) dz (7.32) where H (z ) is some function of z . Here we use a simple second order integration scheme,
the midpoint rule, and write
zm + ∆z
2
zm − ∆z
2 H (z ) dz ≈ H (zm )∆z + O(∆z 2 ) (7.33) 7.5. FINITE VOLUME METHOD FOR SCALAR ADVECTION IN 2D 109 1.5 1 0.5 0
0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 7.11: The midpoint rule approximates the area under the black curve with the
area under the red rectangle. The rectangle height is determined by the function value
at the midpoint of the interval, the dashed red curve. A geometric interpretation of the midpoint rule is shown in ﬁgure 7.11. The area under
the function H (z ), shown in black is approximated by the area under the red triangle,
and whose height is deﬁned by the function value at the midpoint zm . The integration
reduces hence to an evaluation of the function at the edge center multiplied by the size
of the edge, ∆z .
Here we use a simple second order integration scheme, the midpoint rule, and write
yk + ∆ y
2
yk − ∆ y
2 x j + ∆x
2 x j − ∆x
2 F xj + 1 , y dy ≈ F xj + 1 , yk ∆y (7.34) G x, y k + 1 dx ≈ G xj , yk+ 1 ∆x (7.35) 2 2 2 2 The above integration formula are exact if F varies linearly across the cells. Higher order
integration formula could be used but would involve substantially more work. The ﬁnal
form for the 2D FV advection equation becomes:
F xj + 1 , yk − F xj − 1 , yk
dT j,k
2
2
=−
dt ∆y + G xj , yk+ 1 − G xj , yk− 1
2 δA 2 ∆x (7.36)
Equation 7.36 is an ordinary diﬀerential equation governing the time evolution of the
cellaveraged tracer. Its time integration can be performed by one of the timestepping
scheme discussed previously; an appropriate scheme is the third order RungeKutta
method (RK3). The right hand of this ODE requires simply the computations of the
ﬂux divergence onto cell (j, k). The ﬁnal piece missing is the reconstruction of the
function values at the cell edges prior to computing the ﬂuxes, a topic we follow up on
in the next section. 110 CHAPTER 7. FINITE VOLUME METHOD 7.5.1 Function reconstruction in 2D The ﬂux integration requires simply the calculation of:
∆x
, yk
2
∆y
G xj , y k +
2 F xj + ∆x
, yk
2
∆y
= v xj , y k +
2 = Fj + 1 ,k = u xj +
2 = Gj,k+ 1
2 ∆x
, yk
2
∆y
T xj , y k +
2 T xj + (7.37)
(7.38) The scheme requires the evaluation of the T function from its cell averages in neighboring
cells. The 2D geometry now aﬀord a number of cells over which to do the reconstruction.
One can divide ﬁnite volume methods in roughly two categories, one that uses a skewed
stencil to reconstruct the function and those that rely on a straightforward dimensionbydimension reconstruction. It is the latter that will be presented in these notes. Again
1
we will map our cells into a unit cell ξ , η  ≤ 2 according to
ξ= x − xj
y − yk
, η=
∆x
∆y (7.39) where (xj , yk ) is the center of cell (j, k).
Piecewise constant 2D reconstruction
In this case the function T is a constant that does not vary with ξ nor η within the ﬁnite
volume cell. This constant must obviously enough be equal to the cell average, and we
have T = T j,k . The reconstructed function on the cell edges become:
Tj + 1 ,k =
2 T j,k
T j +1,k for
for uj + 1 ,k ≥ 0
2
, Tj,k+ 1 =
2
uj + 1 ,k < 0
2 T j,k
T j,k+1 for
for vj,k+ 1 ≥ 0
2
uj,k+ 1 < 0 (7.40) 2 Piecewise linear 2D reconstruction
In this case the function T is allowed to vary linearly with ξ and η . An obvious candidate
function is
T = a0 + a1 ξ + b1 η
(7.41)
where a0 , a1 and b1 are unknown coeﬃcients that must be determined by imposed conditions. For example by requiring that the cell averages of cells (j, k), (j + 1, k) and
(j, k + 1) be recovered. It is easy then to show that
a0 = T j,k , a1 = T j +1,k − T j,k , b1 = T j,k+1 − T j,k (7.42) This reconstruction is second order. The edge values can now be recovered:
T j,k + T j +1,k
1
Tj + 1 ,k = T (ξ = , η = 0) =
2
2
2
T j,k + T j,k+1
1
Tj,k+ 1 = T (ξ = 0, η = ) =
2
2
2 (7.43)
(7.44) 7.5. FINITE VOLUME METHOD FOR SCALAR ADVECTION IN 2D 111 Piecewise bilinear 2D reconstruction
The linear variation reconstruction proposed in 7.41 is not the only one possible. An
alternative is a bilinear reconstruction:
T = a0 + a1 ξ + b1 η + c1 ξη (7.45) This function varies linearly along constant ξ or η lines, and has a nonlinear (quadratic)
behavior along any other line. Four constraints are now needed to determine the four
constants: the recovery of the cellaverages over cells (j, k), (j + 1, k), (j, k + 1) and
(j + 1, k + 1). The associated system of equation would be
yk + q 2 yk − q 2 xj + p
2 xj − p (a0 + a1 ξ + b1 η + c1 ξη ) dξ dη = T j +p,k+q , p, q = 0, 1 (7.46) 2 This leads to the matrix equation and solution: 1
1
1
1 0
1
0
1 0
0
1
1 0
0
0
1 T j,k
T j +1,k
T j,k+1
T j +1,k+1 a0
a1
b1
c1 1
0
0
−1 1
0
−1 0
1
1 −1 −1 0
0
0
1 T j,k T j +1,k
, = = T j,k+1 T j +1,k+1
(7.47)
It is now a simple matter of evaluating the function at the ﬂux integration points:
a0
a1
b1
c1 1
, η = 0) = a0 +
2
1
= T (ξ = 0, η = ) = a0 +
2 Tj + 1 ,k = T (ξ =
2 Tj,k+ 1
2 T j,k + T j +1,k
1
a1 =
2
2
T j,k + T j,k+1
1
b1 =
2
2 (7.48)
(7.49) Equations 7.49 and 7.44 are identical at the center of cell edges, (actually they coincide
along the entire line ξ = 0 and η = 0) but are diﬀerent at other points. For example at
1
the corner of the cell (ξ, η ) = ( 2 , 1 ) the linear interpolation of 7.41 yields
2
Tj + 1 ,k+ 1 =
2 2 T j +1,k + T j,k+1
2 (7.50) whereas the bilinear interpolation of 7.45 yields
Tj + 1 ,k+ 1 =
2 2 T j,k + T j +1,k + T j,k+1 + T j +1,k+1
4
(7.51) Both linear and bilinear interpolations are valid and yields second order accuracy in the
grid spacing. The bilinear interpolation include information from corner neighbors in its
reconstruction. However, this information is not used when the ﬂux is approximated only
with a single midpoint evaluation at the cell edges. For this information to ﬂow through
the algorithm we need to modify the ﬂux integral and evaluate it more accurately. This
would be useful to do when the ﬂow is not aligned with grid lines. 112 CHAPTER 7. FINITE VOLUME METHOD
Q
2
3
4 z√
q
±√33
15
±5
0
±
± ωq
1 ξ√
q
±√63
15
±6
0 5
9
8
9 √
15+2 30
35
√
15−2 30
35 1
±2 0.347854845137454 1
±2 0.652145154862546 √
15+2 30
35
√
15−2 30
35 ωq
1
2
5
18
8
18 0.173927422568727
0.326072577431273 Table 7.1: Table of GaussLobatto quadrature points. Columns 23 shows the values for
1
the standard interval, while columns 34 shows the values for the interval ξ  ≤ 2 . Higher order 2D reconstruction
Higherorder reconstruction in multidimensions are possible, however, they require a
substantial increase in the amount of computations and coding. The ﬁrst has to do with
an increased complexity of the reconstruction polynomials. This is relatively easy to handle as it could simply be done by a dimension by dimension approach. The additional
complication is to replace the midpoint integration rule, equation 7.35, by something
more accurate. One example is replacing the midpoint rule by Gauss quadrature formula:
Q 1
−1 f (z ) dz ≈ f (zq )ωq (7.52) q =1 where zq are the quadrature points, the ωq are the quadrature weights, and Q is the
number of quadrature points. The approximation is exact if f (z ) is a polynomial of
degree 2Q − 1 in z . Gauss quadrature roots and weights are shown on the standard
1
interval [−1, 1] and [− 2 , 1 ] in table 7.1. Focussing on the integrals of the xﬂuxes on the
2
right edge of the cells, we need to
yk + ∆ y
2
yk − ∆ y
2
x j + ∆x
2
x j − ∆x
2 F xj + 1 , y
2 G x, y k + 1
2 d y = ∆y dx = ∆x 1
2 −1
2
1
2
1
−2 1
F ξ = ,η
2
1
G ξ, η =
2 Q dη ≈ ∆y F
q =1 1
, ηq ωq
2 (7.53) 1
ωq
2 (7.54) Q dξ ≈ ∆x G ξq ,
q =1 These Gauss quadrature formula require the evaluation of the ﬂux at multiple points
along the edges, hence the reconstruction should be capable of providing value for T ( 1 , ηq )
2
and T (ξq , 1 ). Note also that the advective velocity need to available at these points.
2 7.6 Algorithm Summary We are now in a position to summarize the solution process.
1. Read or Set the physical and numerical parameters of the problem. 7.7. CODE DESIGN 113 2. Deﬁne the domain’s geometry including the number of cells in the x, y directions,
the grid sizes, and the cell areas. It would helpfull also to store the coordinates of
the cell centers as well as that of the cell edges.
3. Deﬁne the ﬂow ﬁeld. The code for the Stommel gyre will be provided.
4. Deﬁne the output units, and the output format of the ﬁles, and compute some
preliminary diagnostics, like the budget of T over the domain.
0
5. Deﬁne the initial distribution of the tracer Tj,k . The cell averages at the initial time
can be also deduced. Here and keeping with the second order accuracy philosophy,
we will take the cell averages to be the value of T at the center of the cell. It
is easy to convince yourself using a twodimensional version of the midpoint rule
that T j,k = T (xj , yk ) + O(∆x2 , ∆y 2 ). 6. Start a time loop that call a subroutine to perform a single time integration using
the RK3 routine.
7. The RK3 routine calls a function to compute the right hand side of the ODE,
equation 7.36. It needs to call three times for the 3 stages of the scheme. The right
hand side should accept the cell averages and return the ﬂux divergence.
8. The right hand side computation requires the evaluation of the ﬂuxes at cell edges
prior to performing the ﬂuxes. These are obtained from reconstructing the function
values at the cell edges and multiplying by the velocity ﬁeld.
9. Output some diagnostics, like the T budget within the domain, and the extrema
of the ﬁeld. The solution needs to be saved intermitently for examination. 7.7 Code Design You should start with the onedimensional ﬁnite volume code that was presented in class
as it already includes many of the elements listed above. What is required is to transform
the code from 1D to 2D. One sequence of modiﬁcation is as follows: 7.7.1 Data Structure It is obvious that the most eﬃcient way to store the various data is in two dimensional
matrices. The dimensions of the matrices diﬀer, the cell averages must be Tb(M,N) where
M, N are the number of cells in the x, y directions respectively. 7.7.2 Domain Geometry The domain description needs to be upgraded to account for the multidimensionality
of the problem. The grid information is contained in module grid.f90. The following
information requires to be added to it:
• The grid size in the y direction. 114 CHAPTER 7. FINITE VOLUME METHOD
• The ycoordinate of the cell edges: yk+ 1 = (k − 1)∆y , with k = 1, 2, . . . , N + 1,
2
where N is the number of cells in the ydirections. The y coordinates of the cell
centers may also be needed and should be stored. 7.7.3 Flow The ﬂow ﬁeld information should be updated. The u velocity should be made twodimensional and the y component of the velocity should also be added. Notice that
u need to be deﬁned at vertical cell edges, and consequently its dimension should be
declared as u(M+1,N) whereas v should be declared v(M,N+1). Refer to the ﬁgure for
further information about the grid. 7.7.4 T initiations The setting of the initial condition was done in a subroutine included in module params.f90
that contains all the problem statements. This subroutine should be made twodimensional,
and be passed the proper data in its argument list or via the modules it includes. 7.8 Tracer Advection in a Stommel Gyre Here, we revisit the Hecht advection test problem (???) to characterize the behavior of
the highorder scheme in the underresolved regime. The Hecht advection test consists
of advecting a passive tracer in a Stommel Gyre, and in particular through an intense
westen boundary current where the smooth looking Gaussian Hill undergoes intense
deformation. Underresolved features in this regime produce substantial noise in the
solution and fail to propagate the solution downstream and out of the boundary current
region where the Hill reconstitute itself. 7.8.1 The ﬂow ﬁeld The ﬂow ﬁeld is given by the socalled Stommel gyre model which is an idealized version
of the Gulf Stream system. The ﬂow is characterized by an intense Western Boundary
current that moves waters northward in a narrow zone whose width depends on the β eﬀect and the bottom friction value. The ﬂow in the rest of the domain is slow and has
little shear. The streamline for this ﬂow is given by:
yπ
P eAx + (1 − P )eBx − 1
(7.55)
ψ = Ψ sin
b
1 − eBa
(7.56)
P = Aa
e − eBa where a and b are the zonal and meriodinal size of the basin, respectively (the western
and southern boundaries are located at x = 0 and y = 0, respectively). The parameters
A and B determine the width of the western boundary current region and these, in turn,
depend on the ratio of the β parameter, β , and drag coeﬃcient, Cd as follows:
α= β
Cd (7.57) 7.8. TRACER ADVECTION IN A STOMMEL GYRE 115 6000
5000
4000
3000
2000
1000
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 7.12: Streamlines of Stommel gyre ﬂow ﬁeld. The ﬂow is characterized by an
intense Western Boundary Current and a slow moving southerward current elsewhere in
the basin. The bull’s eye in the lower left corner is the initial distribution of T .
α
+
2 α
2 2 A=− α
−
2 α
2 2 B=− π
b 2 + π
b 2 + (7.58)
(7.59) The strength of the transport depends on the windstress, τ , the water density, ρ, the
drag coeﬃcient and the meridional size of the basin:
Ψ= τπ
ρCd b b
π 2 (7.60) The advective velocity components can be obtained by diﬀerentiating the streamfunction
and dividing by the depth of the basin D
u= 7.8.2 ψx
ψy
, v=−
D
D (7.61) The initial condition The initial condition for our tracer which we can think of as pollutant dumped in the
ocean is a circular Gaussian hill distribution with a decay length scale of l
T = exp − (x − xc )2 + (y − yc )2
l2 and the center of the proﬁle is located at (xc , yc ) = ab
3, 3 . (7.62) 116 CHAPTER 7. FINITE VOLUME METHOD
6000
5000
4000
3000
2000
1000 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000 6000
5000
4000
3000
2000
1000 6000
5000
4000
3000
2000
1000 6000
5000
4000
3000
2000
1000 Figure 7.13: Solution by a method of characteristics that tracks the solution after 1/2
(ﬁrst), 1 (second), 2 (third) and 3 (fourth) years of integration. 7.8. TRACER ADVECTION IN A STOMMEL GYRE 7.8.3 117 Expected result The Gaussian hill will slowly makes its way to the western boundary region where it
will be sheared and deformed substantially. The ﬁdelity of the simulation will depend
crucially on the spatial resolution used. The western boundary current with the current
parameter settings will have a width of about 70 km. If the grid spacing is not enough
to resolve the shear region, numerical noise will be generated and will manifest itself
in either large positive and negative values outside the bounds of the initial condition
(0 ≤ T ≤ 1). One of the aim of this exercise is to decide at what grid resolution,
compared the width of the western boundary current, can one consider the simulation
to be adequate. The report should thus include multiple runs with ∆x = 100, 50, 20, 10
km.
The integration should be carried out for 3 (1080 days or years taking snapshots
every month to record the timeevolution of T . The maximum speed is about 1.5 m/s;
the time step should be scaled accordingly so that the Courant number, C = u∆t/Dx
does not exceed 1/2.
The solution after 3 years of integration is shown in ﬁgure 7.13. 7.8.4 Support Code A number of matlab scripts will be provided to help in visualizing the results in matlab.
The scripts are located in directory ~mohamed/Project on metoﬁs. Feel free to modify
them to ﬁt your needs. 118 CHAPTER 7. FINITE VOLUME METHOD Chapter 8 Numerical Dispersion of
Linearized SWE
This chapter is concerned with the impact of FDA and variable staggering on the ﬁdelity
of wave propagation in numerical models. We will use the shallow water equations as the
model equations on which to compare various approximations. These equations are the
simplest for describing wave motions in the ocean and atmosphere, and they are simple
enough to be tractable with pencil and paper. By comparing the dispersion relation of
the continuous and discrete systems, we can decide which scales of motions are faithfully
represented in the model, and which are distorted. Conversely the diagrams produced
can be used to decide on the number of points required to resolve speciﬁc wavelengths.
The two classes of wave motions encountered here are inertiagravity waves, and Rossby
waves. The main reference for the present work is Dukowicz (1995).
The plan is to look at dynamical system of increasing complexity in order to highlight
various aspects of the discrete systems. We start by looking at 1D versions of the
linearized shallow water equations, and unstaggered and staggered versions of the discrete
approximation; in particular we constrast these two approaches for several high order
centered diﬀerence scheme and show the superiority of the staggered system. Second
we look at the impact of including a second spatial dimensional and include rotation
but restrict ourselves to second order schemes; the discussion is instead focussed on the
various staggering on the dispersive properties. Lastly we look at the dispersive relation
for the Rossby waves. 8.1 Linearized SWE in 1D Since we are interested in applying Fourier analysis to study wave propagations, we need
to linearize the equations and hold the coeﬃcients of the PDE to be constant. For the
shallow water equations, they are:
ut + gηx = 0 (8.1) ηt + Hux = 0 (8.2) 119 120 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE 8.1.1 Centered FDA on Agrid The straight forward approach to discretizing the shallow water equation in space is
to replace the continuous partial derivatives by their discrete counterparts. The main
question is what impact do the choice of variable staggering have on the dispersion
relationship. We start by looking at the case where u and η are colocated. We also
restrict ourselves at centered approximation to the spatial derivatives which have the
form:
M
αm (uj +m − uj −m )
ux j = m=1
+ O(∆x2M )
(8.3)
2∆x
where M is the width of the stencil; the αm are coeﬃcients that can be obtained from the
Taylor series expansions (see equations 3.24,3.27, and 3.29. The order of the neglected
term on an equally spaced grid is O(∆x)2M . A similar representation holds for the η
derivative. The semidiscrete form of the equation is then:
M
m=1 αm (ηj +m ut j + g − ηj −m ) 2∆x
M
αm (uj +m − uj −m )
m=1
2∆x ηt j + H =0 (8.4) =0 (8.5) To compute the numerical dispersion associated with the spatially discrete system,
we need to look at periodic solution in space and time, and thus we set
uj
ηj u
ˆ
η
ˆ = ei(kxj −σt) (8.6) Hence we have for the time derivative ut = −σ uei(kxj −σt) , and for the spatially discrete
ˆ
derivative
uj +m − uj −m = u ei(kxj+m −σt) − ei(kxj−m −σt) = uei(kxj −σt) 2i sin mk∆x;
ˆ
ˆ (8.7) Hence the FDA of the spatial derivative has the following expression
M M
i(kxj −σt) m=1 αm (uj +m − uj −m ) = 2i u e
ˆ αm sin mk∆x (8.8) m=1 Similar expressions can be written for the η variables. Inserting the periodic solutions in
8.5 we get the homogeneous system of equations for the amplitudes u and η :
ˆ
ˆ
M − iσ u + gi
ˆ αm η=0
ˆ (8.9) sin mk∆x
u − iσ η = 0
ˆ
ˆ
∆x (8.10) m=1 M αm Hi
m=1 sin mk∆x
∆x For nontrivial solution we require the determinant of the system to be equal to zero, a
condition that yields the following dispersion relation:
M σ = ±c αm
m=1 sin mk∆x
∆x (8.11) 8.1. LINEARIZED SWE IN 1D
where c =
then √ 121 gH is the gravity wave speed of the continuous system. The phase speed is
M αm CA,M = c
m=1 sin mk∆x
k ∆x (8.12) and clearly the departure of the term in bracket from unity determines the FDA phase
ﬁdelity of a given order M. We thus have the following relations for schemes of order 2,
4 and 6:
sin k∆x
CA,2 = c
(8.13)
k ∆x
8 sin k∆x − sin 2k∆x
CA,4 = c
(8.14)
6k∆x
45 sin k∆x − 9 sin 2k∆x + sin 3k∆x
CA,6 = c
(8.15)
30k∆x 8.1.2 Centered FDA on Cgrid When the variables are staggered on a Cgrid, the spatially discrete equations take the
following form
ut  j + 1 + g M
m=0 βm 2 ηj + 1 + 1+2m − ηj + 1 − 1+2m
2 2 2 2 ∆x
M
βm (uj + 1+2m − uj − 1+2m )
m=0 =0 (8.16) 2
2
=0
(8.17)
∆x
where βm ’s are the diﬀerentiation coeﬃcients on a staggered grid. These can be obtained
from applying the expansion in 3.24 to grids of spacing (2∗m+1)∆x/2 with m = 0, 1, 2, . . .
to get:
uj + 1 − uj − 1
∂u
(∆x/2)2 ∂ 3 u (∆x/2)4 ∂ 5 u (∆x/2)6 ∂ 7 u
2
2
=
+
+
+
(8.18)
∆x
xj
3!
∂x3
5!
∂x5
7!
∂x7
uj + 3 − uj − 3
(3∆x/2)2 ∂ 3 u (3∆x/2)4 ∂ 5 u (3∆x/2)6 ∂ 7 u
∂u
2
2
+
=
+
+
(8.19)
3∆x
xj
3!
∂x3
5!
∂x5
7!
∂x7
uj + 5 − uj − 5
∂u
(5∆x/2)2 ∂ 3 u (5∆x/2)4 ∂ 5 u (5∆x/2)6 ∂ 7 u
2
2
=
+
+
+
(8.20)
5∆x
xj
3!
∂x3
5!
∂x5
7!
∂x7 ηt j + H The fourth order approximation can be obtained by combining these expressions to yield:
1
3
1 uj + 2 − uj − 3
∂u
(∆x/2)4 ∂ 5 u
(∆x/2)6 ∂ 7 u
9 uj + 2 − uj − 1
2
2
−
=
−9
− 90
(8.21)
8
∆x
8
3∆x
xj
5!
∂x5
7!
∂x7
1
1
5
25 uj + 2 − uj − 2
1 uj + 2 − uj − 5
∂u
(∆x/2)4 ∂ 5 u
(∆x/2)6 ∂ 7 u
2
−
=
− 25
− 650
(8.22)
24
∆x
24
5∆x
xj
5!
∂x5
7!
∂x7
Finally the above two expressions can be combined to yield the sixthorder approximation:
5
25 uj + 3 − uj − 3
9 uj + 2 − uj − 5
∂u
(∆x/2)6 ∂ 7 u
450 uj + 1 − uj − 1
2
2
2
2
2
−
+
=
+ 150
384
∆x
128
3∆x
384
5∆x
xj
7!
∂x7
(8.23) 122 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
Going back to the dispersion relation, we now look for periodic solution of the form:
i(kxj + 1 −σt) uj + 1 = ue
ˆ 2 2 and ηj = η ei(kxj −σt)
ˆ (8.24) which when replaced in the FDA yields the following
M M
m=0 βm (uj + 1+2m − uj − 1+2m ) = uei(kxj −σt)
ˆ
2 2 βm ei
m=0
M = uei(kxj −σt) 2i
ˆ
M
m=0 βm sin i k xj + 1 −σt ˆ
βm (ηj + 1 + 1+2m − ηj + 1 − 1+2m ) = η e
2 2 2 m=0
M 2 i k xj + 1 −σt
2 1+2m
k ∆x
2 m=0
M 2i − e−i 1+2m
k ∆x
2 (8.25) 1 + 2m
k ∆x
2 βm ei 2 = ηe
ˆ 1+2m
k ∆x
2 βm sin
m=0 − e−i (8.26)
1+2m
k ∆x
2 1 + 2m
k ∆x
2 (8.27) (8.28) Inserting these expressions in the FDA for the Cgrid we obtain the following equations
after eliminating the exponential factors to get the dispersion equations:
M − iσ u + g2i
ˆ βm
m=0 M βm H 2i
m=1 sin 1+2m k∆x
2
∆x sin 1+2m k∆x
2
∆x η=0
ˆ (8.29) u − iσ η = 0
ˆ
ˆ (8.30) The frequency and the phase speed are then given by
M σC,M = ±c βm
m=0 M
sin 1+2m k∆x
sin 1+2m k∆x
2
2
σC,M = ±c
βm
∆x/2
k∆x/2
m=0 (8.31) We thus have the following phase speeds for schemes of order 2,4 and 6
σC,2 = c
σC,4 = c
σC,6 = c sin k∆x
2
k∆x/2 (8.32) 27 sin k∆x − sin 3 k∆x
2
2
24k∆x/2
2250 sin k ∆x
2 − 125 sin 3 k∆x + 9 sin 5 k∆x
2
2
1920k∆x/2 (8.33)
(8.34) Figure 8.1 compares the shallow water phase speed for the staggered and unstaggered
conﬁguration for various order of centered diﬀerence approximations. The unstaggered
schemes display a familiar pattern: by increasing order the phase speed in the intermediate wavelengths is improved but there is a rapid deterioration for the marginally resolved
waves k∆x ≥ 0.6. The staggered scheme on the other hand displays a more accurate
representation of the phase speed for the entire spectrum. Notice that the second order 8.1. LINEARIZED SWE IN 1D 123 1
0.9
C−6 0.8 C−4
A−2 0.7 A−4
C−2 Cnum/C 0.6
0.5
0.4
A−6
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5
k∆ x/π 0.6 0.7 0.8 Figure 8.1: Phase speed of the spatially discrete linearized shallow water equation. The
solid lines show the phase speed for the Agrid conﬁguration for centered schemes of order
2, 4 and 6, while the dashed lines show the phase speed of the staggered conﬁguration
for orders 2, 4 and 6. 0.9 1 124 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE u, v, η
r r u, v, η AGrid u, v, η
r r u, v, η u, v
r r BGrid
qηj,k u, v u, v
r r u, v bψ CGrid
v
r bψ v
r ur qηj,k bψ ur bψ vr DGrid
u
r qηj,k v r u
r Figure 8.2: Conﬁguration of 4 Arakawa grids
staggered approximation provides a phase ﬁdelity which is comparable to the the fourth
order approximation in the intermediate wavelengths 0.2π ≤ k∆x ≤ 0.6π and superior
for wavelengths k∆x ≥ 0.6. Finally, and most importantly the unstaggered scheme possess a null mode where C = 0 which could manifest itself as a nonpropagating 2∆x
spurious mode; the staggered schemes do not have a null mode. 8.2 TwoDimensional SWE Here we carry out the dispersion relationship for the twodimensional shallow water
equaions in the presence of rotation. We shall consider the two cases of ﬂow on an f plane and ﬂow on a β plane. We will also consider various grid information that include
the Arakawa grids A, B, C, and D. 8.2.1 Inertia gravity waves The linearized equations are given by
ut − f v + gηx = 0 (8.35) vt + f u + gηy = 0 (8.36) ηt + H (ux + vy ) = 0 (8.37) Assuming periodic solutions in time and space of the form
(u, v, η ) = (ˆ, v , η )ei(kx+ly−ωt) ,
uˆˆ
where (k, l) are wavenumbers in the x − y directions, we obtain the following eigenvalue
problem for the frequency σ :
−iσ −f gik
f −iσ
gil
iHk iHl −iσ = −iω −ω 2 + f 2 + c2 (k2 + l2 ) = 0 (8.38) √
Here c = gH is the gravity wave speed. The noninertial roots can be written in the
following form:
σ 2 = 1 + a2 (kd)2 + (ld)2
(8.39) 8.3. ROSSBY WAVES 125 where σ = ω/f is a nondimensional frequency, d is the grid spacing assumed uniform in
both directions and a is the ratio of the Rossby number to the grid spacing, i.e. it is the
number of points per rossby radius:
a= gH
Ro
=
.
d
fd (8.40) Although the continuous dispersion does not depend on a grid spacing, it is useful to
write it in the above form for comparison with the numerical dispersion relations. The
numerical dispersion for the various grids are given by Dukowicz (1995)
2
σA = 1 + a2 sin2 kd + sin2 ld (8.41) 2
σB (8.42) 2
σC
2
σD 8.3 2 = 1 + 2a [1 − cos kd cos ld]
ld
ld
kd
kd
cos2 + 4a2 sin2
+ sin2
= cos2
2
2
2
2
ld
ld
kd
kd
cos2 + a2 cos2
sin2 ld + sin2 kd cos2
= cos2
2
2
2
2 (8.43)
(8.44) Rossby waves The Rossby dispersion relation are given by
σ = −a2 kd 1 + a2 (kd)2 + (ld)2 −1 (8.45) σA = −a2 sin kd cos ld 1 + a2 sin2 kd + sin2 ld
σB = −a2 sin kd 1 + 2a2 [1 − cos kd cos ld]
σC
σD −1 ; (8.46) −1 ld
ld
kd
kd
ld
cos2 + 4a2 sin2
+ sin2
cos2
2
2
2
2
2
−1
ld
ld
kd
= −a2 sin kd cos2
+ sin2
1 + 4a2 sin2
;
2
2
2 = −a2 sin kd cos2 (8.47)
−1 (8.48)
(8.49) where the frequency is now normalized by βd.
The normalized Rossby wave frequencies, and their relative error for the various grids are displayed in ﬁgures 8.88.12 for
various Rossby radius parameters a. Since contour plots are hard to read we also supply
lineplots for special l values l = 0 and l = k in ﬁgure 8.13. From these plots one can
conclude the following:
1. All grid conﬁgurations have a null mode at k∆x = π
2. The C and D grids have a null mode for all zonal wavenumber when ld = π .
3. for a ≥ 2 the B,C and D grids perform similarly for the resolved portion of the
spectrum kd ≤ 2π/5. 126 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
35
1
30
A−Grid B−Grid 0.6 Exact
25 l∆ x/π l∆ x/π 0.8 0.4 20 0.2
0
1 15 l∆ x/π 0.8 C−Grid D−Grid 0.6 10 0.4
5 0.2
0 0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 0 1 0.8 A−Grid σG
1 −σ , r=8 B−Grid 1
l∆ x/π 0.6
0.9
0.4
0.8
0.2 0.7 0
1 0.6 0.8 0.5 C−Grid 0.4 l∆ x/π 0.6 0.3
0.4
0.2
0.2
0.1
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 Figure 8.3: Comparison of the dispersion relation on the Arakawa A, B, C and D grids.
The top ﬁgure shows the dispersion relation while the bottom one shows the relative
error. The parameter a=8. 8.3. ROSSBY WAVES 127
18 16 1
A−Grid B−Grid 0.6 Exact 14 l∆ x/π l∆ x/π 0.8 0.4 12 10 0.2
0
1 8 0.8 C−Grid D−Grid l∆ x/π 6
0.6
4 0.4
0.2
0 2
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 0 1 0.8 A−Grid σG
1 −σ , r=4 B−Grid 1
l∆ x/π 0.6
0.9
0.4
0.8
0.2 0.7 0
1 0.6 0.8 0.5 C−Grid 0.4 l∆ x/π 0.6 0.3
0.4
0.2
0.2
0.1
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.4: Same as 8.3 but for a=4. 1 0 128 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
9 8 1
A−Grid B−Grid 0.6 Exact 7 l∆ x/π l∆ x/π 0.8 0.4 6 0.2 5 0
1 4 0.8 C−Grid D−Grid l∆ x/π 3
0.6
2 0.4
0.2
0 1
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 0 1 0.8 A−Grid σG
1 −σ , r=2 B−Grid 1
l∆ x/π 0.6
0.9
0.4
0.8
0.2 0.7 0
1 0.6 0.8 0.5 C−Grid 0.4 l∆ x/π 0.6 0.3
0.4
0.2
0.2
0.1
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.5: Same as 8.3 but for a=2. 1 0 8.3. ROSSBY WAVES 129
5 4.5 1
A−Grid B−Grid 0.6 4 Exact 3.5 l∆ x/π l∆ x/π 0.8 0.4 3
0.2
2.5
0
1
2 l∆ x/π 0.8 C−Grid D−Grid
1.5 0.6
0.4 1 0.2
0 0.5
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 0 1 0.8 A−Grid σG
1 −σ , r=1 B−Grid 1
l∆ x/π 0.6
0.9
0.4
0.8
0.2 0.7 0
1 0.6 0.8 0.5 C−Grid 0.4 l∆ x/π 0.6 0.3
0.4
0.2
0.2
0.1
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.6: Same as 8.3 but for a=1. 1 0 130 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
3 1
2.5
A−Grid B−Grid 0.6 Exact
l∆ x/π l∆ x/π 0.8 0.4 2 0.2
1.5
0
1
0.8 C−Grid D−Grid l∆ x/π 1
0.6
0.4
0.5
0.2
0 0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 0 1 0.8 A−Grid σG
1 −σ , r=0.5 B−Grid 1
l∆ x/π 0.6
0.9
0.4
0.8
0.2 0.7 0
1 0.6 0.8 0.5 C−Grid 0.4 l∆ x/π 0.6 0.3
0.4
0.2
0.2
0.1
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.7: Same as 8.3 but for a=1/2. 1 0 8.3. ROSSBY WAVES 131
4 1 3 0.8 A−Grid B−Grid Exact
l∆ x/π l∆ x/π 2
0.6
0.4 1 0.2
0
0
1 l∆ x/π 0.8 C−Grid −1 D−Grid 0.6
−2
0.4
−3 0.2
0 0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 −4 1 σG
Rossby 1 −σ 0.8 A−Grid B−Grid r=8
1 l∆ x/π 0.6
0.8
0.4
0.6
0.2 0.4 0
1 0.2 0.8 C−Grid 0 D−Grid −0.2 l∆ x/π 0.6 −0.4
0.4
−0.6
0.2
−0.8
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 1 −1 Figure 8.8: Comparison of Rossby wave dispersion for the diﬀerent Arakawa grids. The
top ﬁgures shows the dispersion while the bottom ones show the relative error. Here
a=8. 132 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
2 1 1.5 0.8 A−Grid B−Grid Exact
l∆ x/π l∆ x/π 1
0.6
0.4 0.5 0.2
0
0
1 l∆ x/π 0.8 C−Grid −0.5 D−Grid 0.6
−1
0.4
−1.5 0.2
0 0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 −2 1 σG
Rossby 1 −σ 0.8 A−Grid B−Grid r=4
1 l∆ x/π 0.6
0.8
0.4
0.6
0.2 0.4 0
1 0.2 0.8 C−Grid 0 D−Grid −0.2 l∆ x/π 0.6 −0.4
0.4
−0.6
0.2
−0.8
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.9: Same as ﬁgure 8.8 but for a=4. 1 −1 8.3. ROSSBY WAVES 133
1 0.8 1
A−Grid B−Grid 0.6 Exact 0.4 0.6 0.4 l∆ x/π l∆ x/π 0.8 0.2
0.2
0
0
1
−0.2 l∆ x/π 0.8 C−Grid D−Grid
−0.4 0.6
0.4 −0.6 0.2
0 −0.8
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 −1 1 σG
Rossby 1 −σ 0.8 A−Grid B−Grid r=2
1 l∆ x/π 0.6
0.8
0.4
0.6
0.2 0.4 0
1 0.2 0.8 C−Grid 0 D−Grid −0.2 l∆ x/π 0.6 −0.4
0.4
−0.6
0.2
−0.8
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.10: Same as ﬁgure 8.8 but for a=2. 1 −1 134 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
1 0.8 1
A−Grid B−Grid 0.6 Exact 0.4 0.6 0.4 l∆ x/π l∆ x/π 0.8 0.2
0.2
0
0
1
−0.2 l∆ x/π 0.8 C−Grid D−Grid
−0.4 0.6
0.4 −0.6 0.2
0 −0.8
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 −1 1 σG
Rossby 1 −σ 0.8 A−Grid B−Grid r=1
1 l∆ x/π 0.6
0.8
0.4
0.6
0.2 0.4 0
1 0.2 0.8 C−Grid 0 D−Grid −0.2 l∆ x/π 0.6 −0.4
0.4
−0.6
0.2
−0.8
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.11: Same as ﬁgure 8.8 but for a=1. 1 −1 8.3. ROSSBY WAVES 135
1 0.8 1
A−Grid B−Grid 0.6 Exact 0.4 0.6 0.4 l∆ x/π l∆ x/π 0.8 0.2
0.2
0
0
1
−0.2 l∆ x/π 0.8 C−Grid D−Grid
−0.4 0.6
0.4 −0.6 0.2
0 −0.8
0 0.5
k∆ x/π 10 0.5
k∆ x/π 1 −1 1 σG
Rossby 1 −σ 0.8 A−Grid B−Grid r=0.5
1 l∆ x/π 0.6
0.8
0.4
0.6
0.2 0.4 0
1 0.2 0.8 C−Grid 0 D−Grid −0.2 l∆ x/π 0.6 −0.4
0.4
−0.6
0.2
−0.8
0 0 0.2 0.4
0.6
k∆ x/π 0.8 1 0 0.2 0.4
0.6
k∆ x/π 0.8 Figure 8.12: Same as ﬁgure 8.8 but for a=1/2. 1 −1 136 CHAPTER 8. NUMERICAL DISPERSION OF LINEARIZED SWE
3 0
−0.5 2 −1
1 σ −1.5 0 −2
−2.5 −1 −3
−2 −3.5
−4 0 0.2 0.4 0.6 0.8 1 −3 0 0.2 1
π − k ∆x 0.4
1
− π 0.6 0.8 1 0.8 1 0.8 1 0.8 1 0.8 1 k ∆x 1.5 0 1
−0.5 σ 0.5
0 −1 −0.5
−1.5
−1
−2 0 0.2 0.4 0.6 0.8 1 −1.5 0 0.2 1
π − k ∆x 0.4
1
− 0.6 π k ∆x 0.4
1
π− k ∆x 0.8 0 0.6
−0.2 0.4
0.2 −0.6 −0.2 σ −0.4 0 −0.4 −0.8 −0.6
−1 0 0.2 0.4 0.6 0.8 1 −0.8 0 0.2 1
π − k ∆x 0.6 0.3 0 0.2
−0.1
0.1
0 −0.3 −0.1 σ −0.2 −0.2
−0.4
−0.3
−0.5 0 0.2 0.4 0.6 0.8 1 −0.4 0 0.2 1
π − k ∆x 0.4
1
− π 0.6 k ∆x 0.15 0 0.1
−0.05
0.05
0 −0.15 −0.05 σ −0.1 −0.1
−0.2
−0.15
−0.25 0 0.2 0.4 0.6 1
π − k ∆x 0.8 1 −0.2 0 0.2 0.4
1
− π 0.6 k ∆x Figure 8.13: Rossby wave frequency σ versus k∆x for, from top to bottom a =, 8,4,2, 1
and 1/2. The left ﬁgures show the case l = 0 and the right ﬁgures the case l = k. The
black line refers to the continuous case and the colored line to the A (red), B (blue), C
(green), and D (magenta) grids. Chapter 9 Solving the Poisson Equations
The textbook example of an elliptic equation is the Poisson equation:
∇2 u = f, x ∈ Ω (9.1) subject to appropriate boundary conditions on ∂ Ω, the boundary of the domain. The
right hand side f is a known function. We can approximate the above equation using
standard second order ﬁnite diﬀerences:
uj +1,k − 2uj,k + uj − 1, k uj,k+1 − 2uj,k + uj, k − 1
+
= fj,k
∆2 x
∆2 y (9.2) The ﬁnite diﬀerence representation 9.2 of the Poisson equation results in a coupled system
of algebraic equations that must be solved simultaneously. In matrix notation the system
can be written in the form Ax = b, where x represents the vector of unknowns, b
represents the right hand side, and A the matrix representing the system. Boundary
conditions must be applied prior to solving the system of equations.
5s s s s s k3s s s s s T4 s 2s 1s
1 s
s
s 2 s
s
s 3
j s
s
s 4 E s
s
s 5 Figure 9.1: Finite Diﬀerence Grid for a Poisson equation. 137 138 CHAPTER 9. SOLVING THE POISSON EQUATIONS Example 12 For a square domain divided into 4x4 cells, as shown in ﬁgure 9.1, subject
to Dirichlet boundary conditions on all boundaries, there are 9 unknowns uj,k , with
(j, k) = 1, 2, 3. The ﬁnite diﬀerence equations applied at these points provide us with
the system: −4 1
0
1
0
0
0
0
0
1 −4 1
0
1
0
0
0
0
0
1 −4 1
0
1
0
0
0
1
0
1 −4 1
0
1
0
0
0
1
0
1 −4 1
0
1
0
0
0
1
0
1 −4 1
0
1
0
0
0
1
0
1 −4 1
0
0
0
0
0
1
0
1 −4 1
0
0
0
0
0
1
0
1 −4 u2,2
u3,2
u4,2
u2,3
u3,3
u4,3
u2,4
u3,4
u4,4 f2,2
f3,2
f4,2
f2,3
f3,3
f4,3
f2,4
f3,4
f4,4 u2,1 + u1,2 u3,1 u4,1 0 = ∆
−
0 0 u2,5 u3,5
u4,5 + u5,4
(9.3)
where ∆x = ∆y = ∆. Notice that the system is symmetric, and pentadiagonal (5 nonzero diagonal). This last property precludes the eﬃcient solution of the system using the
eﬃcient tridiagonal solver.
The crux of the work in solving elliptic PDE is the need to update the unknowns
simultaneously by inverting the system Ax = b. The solution methodologies fall under 2
broad categories:
1. Direct solvers: calculate the solution x = A−1 b exactly (up to roundoﬀ errors).
These methods can be further classiﬁed as:
(a) Matrix Methods: are the most general type solvers and work for arbitrary
nonsingular matrices A. They work by factoring the matrix into a lower
and upper triangular matrices that can be easily inverted. The most general
and robust algorithm for this factorization is the Gaussian elimination method
with partial pivoting. For symmetric real system a slightly faster version relies
on the Cholesky algorithm. The main drawback of matrix methods is that
their storage cost CPU cost grow rapidly with the increase of the number of
points. In particular, the CPU cost grows as O(M 3 ), where N is the number
of unknowns. If the grid has N points in each direction, then for a 2D problem
this cost scales like N 6 , and like N 9 for 3D problems.
(b) FFT also referred to as fast solvers. These methods take advantage of the
structure of separable equations to diagonalize the system using Fast Fourier
Transforms. The eﬃciency of the method rests on the fact that FFT costs
grow like N 2 ln N in 2D, a substantial reduction compared to the N 6 cost of
matrix methods.
2. Iterative Methods calculate an approximation to the solution that minimizes
the norm of the residual vector r = b − Ax. There is a large number of iterative
solvers; the most eﬃcient ones are those that have a fast convergence rate and low
CPU cost per iteration. Often times it is necessary to exploit the structure of the 9.1. ITERATIVE METHODS 139 equations to reduce the CPU cost and accelerate convergence. Here we mention a
few of the more common iterative schemes:
(a) Fixed point methods: Jacobi and GaussSeidel Methods)
(b) Multigrid methods
(c) Krylov Method: Preconditioned Conjugate Gradient (PCG) 9.1 Iterative Methods We will discuss mainly the ﬁxed point iterations methods and (maybe) PCG methods. 9.1.1 Jacobi method The solution of the Poisson equation can be viewed as the steady state solution to the
following parabolic equation:
ut = ∇ 2 u − f
(9.4)
At steady state, t → ∞ the left hand side of the equation goes to zero and we recover the
original Poisson equation. The artiﬁce of introducing a pseudotime t is usefull because
we can now use an explicit method to update the unknowns uj,k individually without
solving a system of equation. Using a FTCS approximation we have:
un+1 = un +
j,k
j,k ∆t n
u
+ un−1,k + un +1 + un −1 − 4un − ∆tfj,k
j
j,k
j,k
j,k
∆2 j +1,k (9.5) where we have assumed ∆x = ∆y = ∆. At steady state, after an inﬁnite number of
iteration, the solution satiﬁes
uj,k = uj,k + ∆t
uj +1,k + uj −1,k + uj,k+1 + uj,k−1 − 4un − ∆tfj,k
j,k
∆2 (9.6) Forming the diﬀerence of equations 9.6 and 9.5 we get:
en+1 = en +
j,k
j,k ∆t n
+ en−1,k + en +1 + en −1 − en ,
e
j,k
j,k
j,k
j
∆2 j +1,k (9.7) where en = un − uj,k . Thus, the error will evolve according to the ampliﬁcation factor
j,k
j,k
associated with the FDE 9.7. We note that the initial conditions for our pseudotime are
not important for the convergence analysis; of course we would like to start our initial
guess as close as possible to the right solution. Second, since we are not interested in
the transient solution, the timeaccuracy is not relevant easier. As a matter of fact, we
would like to use the largest time step possible that will lead us to steady state. A
VonNeumann stability analysis shows that the stability limit is ∆t ≤ 1 ∆2 , substituting
4
this time step in the equation we obtain the Jacobi algorithm:
un+1 =
j,k un+1,k + un−1,k + un +1 + un −1 ∆2
j
j
j,k
j,k
−
fj,k
4
4 (9.8) A VonNeumann analysis shows that the ampliﬁcation factor for this method is given 140 CHAPTER 9. SOLVING THE POISSON EQUATIONS 0.8 0.7 0.7 0.6 0.6
λ∆ y/π 1
0.9 0.8 λ∆ y/π 1
0.9 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.1 0 0.2 0.2 0.3 0.4 0.4 0.6 κ∆ x/π 0.5 0.8 0.6 0 1 0.7 0.8 0.9 0.1 0 0.2 0.2 0.4 0.3 0.4 κ∆ x/π 0.6 0.5 0.8 0.6 1 0.7 0.8 0.9 Figure 9.2: Magnitude of the ampliﬁcation factor for the Jacobi method (left) and GaussSeidel method (right) as a function of the (x, y ) Fourier components.
by cos κ∆x + cos κ∆y
(9.9)
2
where (κ, λ) are the wavenumbers in the (x, y ) directions. A plot of G is shown in ﬁgure
9.2. It is clear that the shortest (κ∆x → π ) and longest (κ∆x → 0) error components
are damped the least. The intermediate wavelengths (κ∆x = π/2) are damped most
eﬀectively.
G= 9.1.2 GaussSeidel method A simple modiﬁcation to the GaussSeidel method can improve the storage and convergence rate of the Jacobi method. Note that in Jacobi, only values at the previous
iterations are used to update the solution. An improved algorithm can be obtained if the
most recent value is used. Assuming that we are sweeping through the grid by increased
j and k indeces, then the GaussSeidel method can be written in the form:
+1
un+1,k + un−1,k + un +1 + un+1 1
j,k
j
j,k −
j ∆2
fj,k
(9.10)
4
4
The major advantages of this scheme are that only one time level need be stored (the
values can be updated on the ﬂy), and the convergence rate can be improved substantially
(double the rate of the Jacobi method). The latter point can be quantiﬁed by looking
at the error ampliﬁcation which now takes the form:
un+1
j,k G= = − 1 + cos(α − β )
eiα + eiβ
G2 =
−iα + e−iβ )
4 − (e
9 − 4(cos α + cos β ) + cos(α − β ) (9.11) where α = κ∆x, and β = λ∆y . A plot of G versus wavenumbers is shown in ﬁgure 9.2,
and clearly shows the reduction in the area where G is close to 1. Notice that unlike
the Jacobi method, the smallest wavelengths are damped at the rate of 1/3 at every time
step. The error components that are damped the least are the long ones: α, β → 0. 9.1. ITERATIVE METHODS 141 1 1 0.9 0.9 0.7 0.6 0.6
λ∆ y/π 0.8 0.7 λ∆ y/π 0.8 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.2 0 0.2 0.3 0.4 0.4 κ∆ x/π 0.6 0.5 0.8 0.6 0 1 0.7 0.8 0.1 0 0.2 0.2 0.4 0.3 0.4 κ∆ x/π 0.6 0.5 0.8 0.6 1 0.7 0.8 0.9 Figure 9.3: Magnitude of the ampliﬁcation factor for the GaussSeidel by rows (left)
and GaussSeidel method by rows and columns (right) as a function of the (x, y ) Fourier
components. 9.1.3 Successive Over Relaxation (SOR) method A correction factor can be added to the update of the GaussSeidel method in order to
improve the convergence rate. Let u∗ denote the temporary value obtained from the
j,k
GaussSeidel step; then an improved estimate of the solution is
un+1 = un + ω (u∗ − un )
j,k
j,k
j,k
j,k (9.12) where ω is the correction factor. For ω = 1 we revert to the GaussSeidel update, for
ω < 1 the correction is underrelaxed, and for ω > 1 the correction is overrelaxed. For
convergence, it can be shown that 1 ≤ ω ≤ 2. The optimal ω , ωo , can be quite hard
to compute and depends on the number of points in each direction and the boundary
conditions applied. Analytic values for ωo can be obtained for a Dirichlet problem:
ωo = 2 1− √ 1−β
, β=
β 2
∆x2
cos Nπ 1
−
∆y 2
2
1 + ∆x2
∆y cos Mπ 1 +
− (9.13) where M and N are the number of points in the x and y directions, respectively. 9.1.4 Iteration by Lines A closer examination of the GaussSeidel method in equation 9.10 reveals that an eﬃcient
algorithm, relying on tridiagonal solvers, can be produced if the iteration is changed to:
un+1
j,k = +1
+1
un+1,k + un−1,k + r 2 (un +1 + un+1 1 )
j,k
j,k −
j
j 2(1 + r 2 ) − ∆ x2
fj,k
4 (9.14) where r = ∆x/∆y is the aspect ratio of the grid. Notice that 9.14 has 3 unknowns only
at row j since un+1 1 would be known from either a boundary condition or a previous
j,k − 142 CHAPTER 9. SOLVING THE POISSON EQUATIONS iteration, and un +1 is still lagged in time. Hence a simple tridiagonal solver can be
j,k
used to update the rows onebyone. The ampliﬁcation factor for this variation on the
GaussSeidel method is given by:
G2 = r4
[2(1 + r 2 − cos α)]2 + [2(1 + r 2 − cos α)]2 cos β + r 4 (9.15) A plot of G for r = 1 is shown in ﬁgure 9.3. The areas with small G have expanded
with resepect to those shown in ﬁgure 9.2. In order to symmetrize the iterations along
the two directions, it is natural to follow a sweepbyrow by a sweepbycolumns. The
ampliﬁcation factor for this iteration is shown in the left panel of ﬁgure 9.3 and show a
substantial reduction in error amplitude for all wavelenths except the longest ones.
Example 13 In order to illustrate the eciency of the diﬀerent methods outline above
we solve the following Laplace equation
∇2 u = 0, 0 ≤ x, y ≤ 1 (9.16) u(0, y ) = u(1, y ) = 0 (9.17) u(x, 1) = sin πx (9.18) −16(x− 1 )2
4 u(x, 1) = e sin πx (9.19) We divide the unit square into M × N grid points and we use the following methods:Jacobi, GaussSeidel, SOR, SOR by line in the xdirection, and SOR by line in both
directions. We monitor the convergence history with the rms change in u from one
iteration to the next:
ǫ 2 1 1
(un+1 − un )2 =
jk
M N j,k jk 2 (9.20) The stopping criterion is ǫ 2 < 10−13 , and we limit the maximum number of iterarions
to 7,000. We start all iterations with u = 0 (save for the bc) as an initial guess. The
convergence history is shown in ﬁgure 9.4 for M = N = 65. The Jacobi and GaussSeidel
have similar convergence history except near the end where GaussSeidel is converging
faster. The SOR iterations are the fastest reducing the number of iterations required
by a factor of 100 almost. We have used the optimal relaxation factor since it is was
computable in our case. The SOR iterations are also quite similar showing a slow decrease
of the error in the initial stages but very rapid decrease in the ﬁnal stages. The criteria
for the selection of an iteration algorithm should not rely solely on the algorithm’s rate
of convergence; it should also the operation count needed to complete each iteration.
The convergence history for the above example shows that the 2way line SOR is the
most eﬃcient per iterations. However, table 9.1 shows the total CPU time is cheapest
for the pointSOR. Thus, the overhead of the tridiagonal solver is not compensated by
the higher eﬃciency of the SOR by line iterations. Table 9.1 also shows that, where
applicable, the FFTbased fast solvers are the most eﬃcient and cheapest. 9.1. ITERATIVE METHODS 143 −2 10 −4 10 −6 ε 2 10 −8 10 −10 10 −12 10 −14 10 0 10 1 10 2 3 10
n 4 10 10 Figure 9.4: Convergence history for the Laplace equation. The system of equation is
solved with: Jacobi (green), GaussSeidel (red), SOR (black), Line SOR in x (solid
blue), and line SOR in x and y (dashed blue). Here M = N = 65. Jacobi
GaussSeidel
SOR
SORLine
SORLine 2
FFT 33
0.161
0.131
0.009
0.013
0.014
0.000 65
0.682
2.197
0.056
0.164
0.251
0.001 129
2.769
10.789
0.793
1.291
1.403
0.004 Table 9.1: CPU time in second to solve Laplace equation versus the number of points
(top row). 144 9.1.5 CHAPTER 9. SOLVING THE POISSON EQUATIONS Matrix Analysis The relaxation schemes presented above are not restricted to the Poisson equation but
can be reintrepeted as speciﬁc instances of a larger class of schemes. We present the
matrix approach in order to unify the diﬀerent schemes presented. Let the matrix A be
split into
A=N −P
(9.21)
where N and P are matrices of the same order as A. The system of equations becomes:
Nx = Px + b (9.22) Starting with an arbitrary vector x(0) , we deﬁne a sequence of vectors x(v) by the recursion
N x(n) = P x(n−1) + b, n = 1, 2, 3, ... (9.23) It is now clear what kind of restrictions need to be imposed on the matrices in order to
solve for x, namely: the matrix N must be nonsingular: det(N ) = 0, and the matrix N
must be easily invertible so that computing y from N y = z is computationally eﬃcient.
In order to study how fast the iterations are converging to the correct solution, we
introduce the matrix M = N −1 P , and the error vectors e(n) = x(n) − x. Substracting
equation 9.22 from equation 9.23, we obtain an equation governing the evolution of the
error, thus:
e(n) = M e(n−1) = M 2 e(n−2) = . . . = M n e(0)
(9.24)
where e(0) is the initial error. Thus, it is clear that a suﬃcient condition for convergence,
i.e. that limn→∞ e(n) = 0, is that limn→∞ M n = O. This is also necessary for the
method to converge for all e(0) . The condition for a matrix to be convergent is that its
spectral radius ρ(M ) < 1. (Reminder: the spectral radius of a matrix M is dened as the
maximum eigenvalue in magnitude: ρ(M ) = maxi λi ). Since computing the eigenvalues
is diﬃcult usually, and since the spectral radius is a lower bound for any matrix norm,
we often revert to imposing conditions on the matrix norm to enforce convergence; thus
ρ(M ) ≤ M < 1. (9.25) In particular, it is common to use either the 1 or inﬁnitynorms since these are the
simplest to calculate.
The spectral radius is also useful in deﬁning the rate of convergence of the method.
In fact since, using equation 9.24, one can bound the norm of the error by:
e(n)
(n) e ≤ Mn e(0) (9.26) n (9.27) ≤ [ρ(M )] (0) e Thus the number of iteration needed to reduce the initial error by a factor of α is
n ≥ ln α/ ln[ρ(M )]. Thus, a small spectral radius reduces the number of iterations (and
hence CPU cost) needed for convergence. 9.1. ITERATIVE METHODS 145 Jacobi Method
The Jacobi method derived for the Poisson equation can be generalized by deﬁning the
matrix N as the diagonal of matrix A:
N = D, P = A − D (9.28) The matrix D = aij δij , where δij is the Kronecker delta. The matrix M = D−1 (D − A) =
I − D −1 A In component form the update takes the form:
xn =
i 1
aii K
n
aij xj −1 (9.29) j =1 j =i The procedure can be employed if aii = 0, i.e. all the diagonal elements of A are diﬀerent
from zero. The rate of convergence is in general diﬃcult to obtain since the eigenvalues
are not easily available. However, the inﬁnity and/or 1norm of M can be easily obtained:
ρ(M ) ≤ min( M
M 1 = max j
i=1 1, M ∞)
aij
<1
aii (9.30)
(9.31) i=j M ∞ = max i
j =1 aij
<1
ajj (9.32) j =i (9.33)
GaussSeidel Method
A change of splitting leads to the GaussSeidel method. Thus we split the matrix into a
lower triangular matrix, and an upper triangular matrix: N = a11
a21
.
.
. a22 aK 1 aK 2 .. .
· · · aKK , P = N − A (9.34) A slightly diﬀerent form of writing this splitting is as A = D + L + U where D is again
the diagonal part of A, L is a strictly lower triangular matrix, and U is a strictly upper
triangular matrix; here N = D + L. The matrix notation for the SOR iteration is a little
complicated but can be computed:
xm = M xm−1 + (I + αD−1 L)−1 αD−1 b
M = (I + αD −1 −1 L) [(1 − α)I − αD −1 (9.35)
U] (9.36) 146 9.2 CHAPTER 9. SOLVING THE POISSON EQUATIONS Krylov MethodCG Consider the system of equations Ax = b, where the matrix A is a symmetric positive
deﬁnite matrix. The solution of the system of equations is equivalent to minimizing the
functional:
1
Φ(x) = xT Ax − xT b.
(9.37)
2
The extremum occurs for ∂ Φ = Ax − b = 0, thanks to the symmetry of the matrix, and
∂x
2
the positivity of the matrix shows that this extremum is a minimum, i.e. ∂ Φ = A. The
∂x2
iterations have the form:
xk = xk−1 + αpk
(9.38)
where xk is the kth iterate, α is a scalar and pk are the search directions. The two
parameters at our disposal are α and p. We also deﬁne the residual vector rk = b − Axk .
We can now relate Φ(xk ) to Φ(xk−1 ):
Φ(xk ) = 1T
x Axk − xT b
k
2k = Φ(xk−1 ) + αxT−1 Apk +
k α2 T
p Apk − αpT b
k
2k (9.39) For an eﬃcient iteration algorithm, the 2nd and 3rd terms on the right hand side of
equation 9.39 have to be minimized separately. The task is considerably simpliﬁed if we
require the search directions pk to be Aorthogonal to the solution:
xT−1 Apk = 0.
k (9.40) The remaining task is to choose α such that the last term in 9.39 is minimized. It is a
simple matter to show that the optimal α occurs for
α= pT b
k
,
pT Apk
k (9.41) and that the new value of the functional will be:
Φ(xk ) = Φ(xk−1 ) − 1 (pT b)2
k
.
2 pT Apk
k (9.42) We can use the orthogonality requirement 9.40 to rewrite the above two equations as:
α= pT rk−1
1 (pT rk−1 )2
k
k
, Φ(xk ) = Φ(xk−1 ) −
.
2 pT Apk
pT Apk
k
k (9.43) The remaining task is deﬁning the iteration is to determine the algorithm needed to
update the search vectors pk ; the latter must satisfy the orthogonality condition 9.40,
and must maximum the decrease in the functional. Let us denote by Pk the matrix 9.2. KRYLOV METHODCG 147 formed by the (k − 1) column vectors pi , then since the iterates are linear combinations
of the search vectors,we can write:
xk − 1 =
Pk−1 =
y= k −1
i=1 αi pi = Pk−1 y p 1 p 2 . . . p k −1
α1
α2
.
.
.
αk−1 (9.44)
(9.45) (9.46) We note that the solution vector xk−1 belongs to the space spanned by the search vectors
T
pi , i = 1, . . . , k − 1. The orthogonality property can now be written as y T Pk−1 Apk = 0.
This property is easy to satisfy if the new search vector pk is Aorthogonal to all the
T
previous search vectors, i.e. if Pk−1 Apk = 0. The algorith can now be summarized as
follows: First we initialize our computations by deﬁning our initial guess and its residual;
second we perform the following iterations:
while rk < ǫ:
1. Choose pk such that pT Apk = 0, ∀i < k, and maximize pT rk−1 .
i
k
2. Compute the optimal αk = end p T r k −1
k
pT Apk
k 3. Update the guess xk = xk−1 + αk pk , and residual rk = rk−1 − αk Apk A vector pk which is Aorthogonal to all previous search, and such that pT rk−1 = 0,
k
vectors can always be found. Note that if pT rk−1 = 0, then the functional does not
k
decrease and the minimum has been reached, i.e. the system has been solved. To bring
about the largest decrease in Φ(xk ), we must maximize the inner product pT rk−1 . This
k
can be done by minimizing the angle between the two vectors pk and rk−1 , i.e. minimizing
rk−1 − pk .
Consider the following update for the search direction:
pk = rk−1 − APk−1 zk−1 (9.47) where zk−1 is chosen to minimize J = rk−1 − APk−1 z 2 . It is easy to show that the
minimum occurs for
T
T
Pk−1 aT APk−1 z = Pk−1 AT rk−1 ,
(9.48)
and under this condition pT APk−1 = 0, and pk − rk is minimized. We have the following
k
property:
TT
Pk rk = 0,
(9.49)
i.e. the search vectors are orthogonal to the residual vectors. We note that
Span{p1 , p2 , . . . , pk } = Span{r0 , r1 , . . . , rk−1 } = Span{b, Ab, . . . , Ak−1 b} (9.50) 148 CHAPTER 9. SOLVING THE POISSON EQUATIONS i.e. these diﬀerent basis sets are spanning the same vector space. The ﬁnal steps in the
conjugate gradient algorithm is that the search vectors can be written in the simple form:
pk = rk−1 + βk pk−1 ,
pT Ark−1
k
βk = − T−1
,
pk−1 Apk−1 (9.51) αk = − (9.53) (9.52) T
rk−1 rk−1
pT Apk
k The conjugate gradient algorithm can now be summarized as
Initialize: r = b − Ax, p = r , ρ = r 2 .
while ρ < ǫ:
k ←k+1
w ← Ap
2
α = pr w
T
update guess: x ← x + αp
update residual: r ← r − αw
new residual norm: ρ′ ← r 2
update search direction: β = ρ′ /ρ, p ← r + βp
update residual norm: ρ ← ρ′ .
end
It can be shown that the error in the CG algorithm after k iteration is bounded by:
xk − x A ≤ 2 x0 − x A √
κ−1
√
κ+1 k (9.54) where κ(A) is the condition number,
κ(A) = A A−1 = maxi (λi )
,
mini (λi ) (9.55) the ratio of maximum eigenvalue to minimum eigenvalue. The error estimate uses the
√
Anorm: w A = wT Aw . Note that for very large condition numbers, 1 ≪ κ, the rate
of residual decrease approaches 1:
√
κ−1
√
κ+1 2
≈1− √ ≈1
κ (9.56) Hence the number of iterations needed to reach convergence increases. For eﬃcient
iterations κ must be close to 1, i.e. the eigenvalues cluster around the unit circle. The
˜˜
problem becomes one of converting the original problem Ax = b into Ax = ˜ with
b
˜
κ(A) ≈ 1. 9.3. DIRECT METHODS 149 9.3 Direct Methods 9.3.1 Periodic Problem We will be mainly concerned with FFT based direct methods. These are based on the
eﬃciency of the FFT to diagonalize the matrix A trough the transformation D = Q−1 AQ,
where Q is the unitary matrix made up of the eigenvectors of A. These eigenvector
depend on the shape, boundary conditions of the problem. The method is applicable to
seperable elliptic problems only. For a doubly periodic problem, we can write that:
ujk = M −1 N −1 1
MN umn e−i
ˆ 2πjm
M e−i 2πkn
N (9.57) m=0 n=0 where umn are the Discrete Fourier Coeﬃcients. A similar expression can be written for
ˆ
the right hand side function f . Replace the Fourier expression in the original Laplace
equation we get:
1
MN M −1 N −1 umn e−i
ˆ 2πjm
M e−i 2πkn
N e−i 2πm
M + ei 2πm
M e−i 2πn
N + ei 2πn
N = m=0 n=0 ∆2 1
MN M −1 N −1 2πjm
2πkn
ˆ
fmn e−i M e−i N (9.58) m=0 n=0 Since the Fourier functions form an orthogonal basis, the Fourier coeﬃcients should
match individually. Thus, one can obtain the following expression for the unknowns
umn :
ˆ
umn =
ˆ 9.3.2 ˆ
∆2 fmn
πm
πn
2 cos 2M + cos 2N − 2 , m = 0, 1, . . . , M − 1, n = 0, 1, . . . , N − 1 (9.59) Dirichlet Problem For a Dirichlet problem with homogeneous boundary conditions on all boundaries, the
following expansion satisﬁes the boundary conditions identically:
ujk = 1
MN M −1 N −1
m=1 n=1 umn sin
ˆ πjm
πkn
sin
M
N (9.60) Again, the sine basis function are orthogonal and hence the Fourier coeﬃcients can be
computed as
umn =
ˆ ˆ
∆2 fmn
, m = 1, . . . , M − 1, n = 1, . . . , N − 1
2 cos πm + cos πn − 2
M
N (9.61) Again, the eﬃciency of the method rests on the FFT algorithm. Specialized routines for
sinetransforms are available. 150 CHAPTER 9. SOLVING THE POISSON EQUATIONS Chapter 10 Nonlinear equations
Linear stability analysis is not suﬃcient to establish the stability of ﬁnite diﬀerence
approximation to nonlinear PDE’s. The nonlinearities add a severe complications to
the equations by providing a continuous source for the generation of small scales. Here
we investigate how to approach nonlinear problems, and ways to mitigate/control the
growth of nonlinear instabilities. 10.1 Aliasing In a constant coeﬃcient linear PDE, no new Fourier components are created that are
not present either in the initial and boundary conditions conditions, or in the forcing
functions. This is not the case if nonlinear terms are present or if the coeﬃcients of a
linear PDE are not constant. For example, if two periodic functions: φ = eik1 xj and
ψ = eik2 xj , are multiplied during the course of a calculation, a new Fourier mode with
wavenumber k1 + k2 is generated:
φψ = ei(k1 +k2 )xj . (10.1) The new wave generated will be shorter then its parents if k1,2 have the same sign,
i.e. k12πk2 < k2π2 . The representation of this new wave on the ﬁnite diﬀerence grid can
+
1,
become problematic if its wavelength is smaller then twice the grid spacing. In this case
the wave can be mistaken for a longer wave via aliasing.
Aliasing occurs because a function deﬁned on a discrete grid has a limit on the shortest
wave number it can represent; all wavenumbers shorter then this limit appear as a long
wave. The shortest wavelength representable of a ﬁnite diﬀerence grid with step size ∆x
is λs = 2∆x and hence the largest wavenumber is kmax = 2π/λs = π/∆x. Figure 10.1
shows an example of a long and short waves aliased on a ﬁnite diﬀerence grid consisting
6πx
of 6 cells. The solid line represents the function sin 4∆x and is indistinguishable from
πx
the function sin 4∆x (dashed line): the two functions coincide at all points of the grid
(as marked by the solid circles). This coincidence can be explained by rewriting each
Fourier mode as:
eikxj = eikj ∆x = eikj ∆x ei2πn
151 j 2πn = ei(k+ ∆x )j ∆x , (10.2) 152 CHAPTER 10. NONLINEAR EQUATIONS 6πx
6πx
Figure 10.1: Aliasing of the function sin 4∆x (solid line) and the function sin 4∆x (dashed
line). The two functions have the same values on the FD grid j ∆x. where n = 0, ±1, ±2, . . . Relation 10.2 is satisﬁed at all the FD grid points xj = j ∆x; it
shows that all waves with wavenumber k + 2πn are indistinguishable on a ﬁnite diﬀerence
∆x
grid with grid size ∆x. In the case shown in ﬁgure 10.1, the long wave has length 4∆x
and the short wave has length 4∆x/3, so that the equation 10.2 applies with n = −1.
Going back to the example of the quadratic nonlinearity φψ , although the individual
functions, φ and ψ , are representable on the FD grid, i.e. k1,2  ≤ π/∆x, their product
may not be since now k1 + k2  ≤ 2π/∆x. In particular, if π/∆x ≤ k1 + k2  ≤ 2π/∆x,
the product will be unresolvable on the discrete grid and will be aliased into wavenumber
˜
k given by
π
2
k1 + k2 − ∆π , if k1 + k2 > ∆x
x
˜
(10.3)
k=
π
2π
k1 + k2 + ∆x , if k1 + k2 < − ∆x
˜
Note that very short waves are aliased to very long waves: k → 0 when k1,2  →
d kc
0 ˜
k
t T π
∆x . t k1 + k2
π
∆x & % The k E 2π
∆x Figure 10.2: Folding of short waves into long waves.
aliasing wavenumber can be visualized by looking at the wavenumber axis shown in ﬁgure
π
˜
10.2; note that k1 + k2 and k are symmetrically located about the point ∆x . There
˜
exist a cutoﬀ wavenumber kc whereby all longer wavelength are aliased to k > kc; thus
since k < kc , then (k1 + k2 ) < 2kc . Thus if k1 + k2 > kmax the product will be aliased 10.2. 1D BURGER EQUATION 153 ˜
˜
into k  = kmax − (2kc − kmax ), and the latter must satisfy k  > kc , and we end up with
2
kc < kmax
3
For a ﬁnite diﬀerence grid this is equivalent to kc < 10.2 (10.4)
2π
3∆x . 1D Burger equation The 1D Burger equation is the simplest model of nonlinearities as found in the NavierStokes momentum equations:
∂u
∂u
+u
= 0,
∂t
∂x 0≤x≤L (10.5) Multiplying the above equation by um where m ≥ 0 we can derive the following conservation law:
1 ∂um+2
∂um+1
+
= 0.
(10.6)
∂t
m + 2 ∂x
The above equation is a conservation equation for all moments of the solution; in particular for m = 0, we have the momentum conservation, and for m = 1 energy conservation.
The spatial integral yields:
∂ m+1 dx
Lu ∂t + um+2 L − um+2 0
= 0.
m+2 (10.7) which shows that the global budget of um+1 depends on the boundary values, and is zero
for periodic boundary conditions.
We will explore the impact of spatial diﬀerencing on the continuum conservation
properties of the energy (second moment). A central diﬀerence scheme of the advective
form of the equation, equation 10.5, yields:
uj +1 − uj −1
∂uj
= − uj
.
∂t
2∆x (10.8) Multiplying by uj and summing over the interval we get:
∂ N
2
j =0 uj /2 ∂t =− 1N
(uj uj +1 − uj uj −1 ).
2 j =0 (10.9) Notice that the terms within the summation sign do not cancel out, and hence energy is
not conserved. Likewise, a ﬁnite diﬀerence approximation to the conservative form:
u2+1 − u2−1
∂uj
j
j
=−
,
∂t
4∆x (10.10) is not conserving as its discrete energy equation attests:
∂ N
2
j =0 uj /2 ∂t =− 1N
(uj u2+1 − uj u2−1 ).
j
j
4 j =0 (10.11) 154 CHAPTER 10. NONLINEAR EQUATIONS 1.5
0.56 1 0.55
0.54 0.5
2 ∆x Σi=1 u /2 0.53 N 0 0.52
0.51 −0.5 0.5 −1
0.49 −1.5
−1 −0.5 0
x 0.5 1 0.48
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t Figure 10.3: Left: Solution of the inviscid Burger equation at t = 0.318 < 1/π using
the advective form (black), momentum conserving form (blue), and energy conserving
form (red); the analytical solution is superimposed in Green. The initial conditions are
u(x, 0) = − sin πx, the boundary conditions are periodic, the time step is ∆t = 0.01325,
and ∆x = 2/16; RK4 was used for the time integration. Right: Energy budget for the
diﬀerent Burger schemes: red is the energy conserving, blue is the momentum conserving,
and black is the advective form.
The only energy conserving available for the Burger equation is the following:
∂uj
uj +1 + uj + uj −1 uj +1 − uj −1
=−
.
∂t
3
2∆x (10.12) where the advection velocity is a threeterm average of the velocity at the central point.
Its discrete energy equation is given by:
∂ N
2
j =0 uj /2 ∂t =− 1N
[uj uj +1 (uj + uj +1 ) − uj uj −1 (uj + uj −1 )]
6 j =0 (10.13) where the term inside the summation sign does cancel out. Figure 10.3 shows solutions
of the Burger equations using the 3 schemes listed above. The advective form, shown
in black, does not conserve energy and exhibits oscillations near the front region. The
oscillations are absent in the both the ﬂux and energy conserving forms. The ﬂux form,
equation 10.10, exhibits a decrease in the energy and a decrease in the amplitude of the
waves. Note that the solution is shown just prior to the formation of the shock at time
t = 0.318 < 1/π . 10.3 Quadratic Conservation It is obvious that building quadratic conserving schemes depends highly on the system
of equations considered. The remaining sections will be devoted to equations commonly 10.3. QUADRATIC CONSERVATION 155 found in the CFD/oceanic literature. We will concentrate on the following system of
equations:
vt + v · ∇v + g∇η = f (10.14) αηt + ∇ · (hv) = 0 (10.15) Equation 10.14 is the momentum equation and 10.15 is the continuity equation in a
ﬂuid moving with velocity v and subject to a pressure η . The parameter α controls the
compressibility of the system; for α = 0 we recover the incompressible equations, and for
α = 1 the shallow water equations. The parameter h is the thickness of the ﬂuid layer,
and f is a term lumping momentum sources and sinks (including dissipation).
The system 10.1410.15 imply conservation laws for energy, vorticity and enstrophy
in the incompressible case, and for energy, potential vorticity and potential enstrophy in
the compressible case, when the source terms on the right hand sides are identically zero.
The question is: Is it possible to enforce the conservation of these higher order quantities
(mainly quadratic in the unknown variables) in the ﬁnite diﬀerence approximation? We
will look in particular at energy/enstrophy conservation in nondivergent shallow water
ﬂow, and on energy/potential vorticity conservation in divergent shallow water equations.
We ﬁrst begin by deﬁning the following operators:
− u(x − ∆x )
2
(10.16)
∆x
u(x + ∆x ) + u(x − ∆x )
2
2
(10.17)
ux =
2
The operators deﬁned above are nothing but the centered diﬀerence and averaging operators. It is easy to show that the following relationships holds for any 2 functions a and
b deﬁned on a ﬁnite diﬀerence grids:
δx u = ∆x
2) u(x + x δx (ax ) = δx a (10.18)
x x a δx b = δx (ab) − b δx a
x (10.19) x aδx b = δx (a b) − bδx a
∆x2
x
x
(δx a) (δx b)
ab = ax b +
4
∆ x2
x
x
ax b = ab + δx
b δx a
4 (10.20)
(10.21)
(10.22) The ﬁrst relationship shows that the diﬀerencing and averaging operators commute, the
second and third relations are the ﬁnite diﬀerence equivalent of product diﬀerentiation
rules, and the fourth and ﬁfth are the ﬁnite diﬀerence equivalent of product averaging.
It is easy to show the additional relationships
ax δx a = δx a2
2 (10.23)
x a2
2 x = δx x a2
= (a ) −
2 aδx a
a2
2 x2 x , a2 = a x + ∆x
2 a x− ∆x
2 (10.24) x (10.25) 156 10.4 CHAPTER 10. NONLINEAR EQUATIONS Nonlinear advection equation The advection equation
DT
= Tt + v · ∇T = 0
Dt (10.26) of a tracer T by a ﬂow ﬁeld v that is divergencefree:
∇ · v = 0, (10.27) is equivalent to the following conservation law:
Tt + ∇ · (vT ) = 0. (10.28) Equation 10.26 is called the advective form, and equation 10.28 is called the ﬂux (or conservative) form. The two forms are equivalent in the continuum provided the ﬂow is divergence free. Note that the above statements holds regardless of the linearity/dimensionality
of the system. Integration of the ﬂux form over the domain Ω shows that
d
dt T dV
Ω =− ∂Ω n · v T dS (10.29) where n is the unit outward normal, ∂ Ω the boundary of the ﬂow domain, dS is an
elemental surface on this boundary, and the boundary integral is the amount of T entering
Ω. Equation 10.29 shows that the total inventory of T and Ω depends on the amount of
T entering through the boundary. In particular the total budget of T should be constant
if the boundary is closed v · n = 0 or if the domain is periodic.
The above conservation laws imply higher order conservation. To wit, equation 10.26,
can be multiplied by T m (where m ≥ 0) and the following equation can be derived:
∂T m+1
+ v · ∇T m+1 = 0,
∂t (10.30) i.e. the conservation law imply that all moments of T m are also conserved. Since the
above equation has the same form as the original one, the total inventory of T m will also
be conserved under the same conditions as equation 10.29. 10.4.1 FD Approximation of the advection term For a general ﬂow ﬁeld, the FD of the advection form will not conserve the ﬁrst moment,
while the FD of the ﬂux form will. This is easy to see since the ﬂux is:
∇ · (vT ) = δx (uT ) + δy (vT ). (10.31) The relevant question is: is it possible to come up with a ﬁnite diﬀerence scheme that will
conserve both the ﬁrst and second moment? Let us look at the following approximation
of the ﬂux form
x
y
(10.32)
∇ · (vT ) = δx (ux T ) + δy (v y T ). 10.4. NONLINEAR ADVECTION EQUATION 157 Can the term T ∇ · (vT ) be also written in ﬂux form (for then it will be conserved upon
summation). We concentrate on the xcomponent for simplicity:
T δx ux T x = δx ux T
x = δx u T x = δx u T = δx ux T
+ x2 x2 x2 x2 x − ux T δx T
− x T2
2 ux δx (10.33)
x (10.34)
x x T2
T2
u
δx u
−
2
2
2x
T2
∆ x2
xT
u
δx δx u δx
−
2
4
2 − δx
− δx T2
∆ x2
δx δx u δx
4
2 (10.35)
+ T2
δx ux
2
(10.36) x T2 x
T2
δx (ux )
u−
2
2 = δx (10.37) Equality 10.33 follows from property 10.20, 10.34 from 10.23, 10.35 from 10.19. The
second and third terms of equation 10.35 can rewritten with the help of equations 10.21
and 10.24, respectively. The third and ﬁfth terms on the right hand side of equation
10.36 cancel. The ﬁnal equation 10.37 is obtained by combining the ﬁrst and second
terms of equation 10.36 (remember the operators are linear), and using equation 10.25.
A similar derivation can be carries out for the y component of the divergence:
y y T δy v T y = δy T2 y
T2
δy (v y )
v−
2
2 (10.38) Thus, the semidiscrete second moment conservation becomes:
y x T2 y
T2
∂T 2 /2
T2 x
u − δy
v+
= −δx
[δx (ux ) + δy (v y )]
∂t
2
2
2 (10.39) The ﬁrst and second term in the semidiscrete conservation equation are in ﬂux form,
and hence will cancel out upon summation. The third term on the right hand side is
nothing but the discrete divergence constraint. Thus, the second order moment of T will
be conserved provided that the velocity ﬁeld is discretely divergencefree.
The following is a FD approximation to v · ∇T consistent with the above derivation:
u ∂T
∂x ∂uT
∂u
−T
∂x
∂x
x
= δx ux T − T δx (ux )
= (10.40)
(10.41) x = T δx (ux ) + ux δx T − T δx (ux ) = ux δx T
Thus, we have x x v · ∇T = ux δx T + v y δy T y (10.42)
(10.43)
(10.44) 158 CHAPTER 10. NONLINEAR EQUATIONS 10.5 Conservation in vorticity streamfunction formulation Nonlinear instabilities can develop if energy is falsely generated and persistently channeled towards the shortest resolvable wavelengths. Arakawa Arakawa (1966); Arakawa and Lamb
(1977) devised an elegant method to eliminate these artiﬁcial sources of energy. His
methodology is based on the streamfunctionvorticity formulation of two dimensional,
divergencefree ﬂuid ﬂows. The continuity constraint can be easily enforced in 2D ﬂow
by introducing a streamfunction, ψ , such that v = k × ∇ψ ; in component form this is:
u = −ψy , v = ψx (10.45) ζ = vx − uy = ψx x + ψy y = ∇2 ψ (10.46) The vorticity ζ = ∇ × v reduces to
and the vorticity advection equation can be obtained by taking the curl of the momentum
equation, thus:
∂ ∇2 ψ
= J (∇2 ψ, ψ )
(10.47)
∂t
where J stand for the Jacobian operator:
J (a, b) = ax by − bx ay (10.48) The Jacobian operator possesses some interesting properties
1. It is antisymmetric, i.e.
J (b, a) = −J (a, b) (10.49) 2. The Jacobian can be written in the useful forms:
J (a, b) = ∇a · k × ∇b (10.50) = ∇ · (k × a∇b) (10.51) = −∇ · (k × b∇a) (10.52) 3. The integral of the Jacobian over a closed domain can be turned into a boundary
integral thanks to the above equations
J (a, b)dA =
Ω a
∂Ω ∂b
ds = −
∂s b
∂Ω ∂a
ds
∂s (10.53) where s is the tangential direction to the boundary. Hence, the integral of the
Jacobian vanishes if either a or b is constant along ∂ Ω. In particular, if the boundary
is a streamline or a vortex line, the Jacobian integral vanishes. The areaaveraged
vorticity is hence conserved.
4. The following relations hold:
a2
, b)
2
b2
bJ (a, b) = J (a, )
2 aJ (a, b) = J ( (10.54)
(10.55) 10.5. CONSERVATION IN VORTICITY STREAMFUNCTION FORMULATION 159
Thus, the area integrals of aJ (a, b) and bJ (a, b) are zero if either a or b are constant
along the boundary.
It is easy to show that enstrophy, ζ 2 /2, and kinetic energy, ∇ψ 2 /2, are conserved if
the boundary is closed. We would like to investigate if we can conserve vorticity, energy
and enstrophy in the discrete equations. We begin ﬁrst by noting that the Jacobian in
the continuum form can be written in one of 3 ways:
J (ζ, ψ ) = ζx ψy − ζy ψx (10.56) = (ζψy )x − (ζψx )y (10.57) = (ψζx )y − (ψζy )x (10.58) We can thus deﬁne 3 centered diﬀerence approximations to the above deﬁnitions:
x y y J1 (ζ, ψ ) = δx ζ δy ψ − δy ζ δx ψ
J2 (ζ, ψ ) = δx ζ δy ψ
J3 (ζ, ψ ) = δy ψ δx ζ yx x (10.59) − δy ζ δx ψ xy − δx ψ δy ζ xy (10.60) yx (10.61) It is obvious that J2 and J3 will conserve vorticity since they are in ﬂux form; J1 can
also be shown to conserve the ﬁrst moment since:
xy x J1 (ζ, ψ ) = δx δy ψ ζ x − ζ δx δy ψ xy x = δx δy ψ ζ − yx xy y − δy δx ψ ζ y + ζ δy δx ψ xy (10.62) ∆x2
∆y 2
y
xy y
x
.
δx δy ψ δx ζ − δy δx ψ ζ −
δy δx ψ δy ζ (10.63)
4
4 The last equation above shows that J1 can indeed be written in ﬂux form, and hence
vorticity conservation is ensured. Now we turn our attention to the conservation of
quadratic quantities, namely, kinetic energy and enstrophy. It is easy to show that J2
conserves kinetic energy since:
ψJ2 (ζ, ψ ) = ψδx ζ δy ψ yx x yx x yx y xy = δx ψ ζδy ψ
= δx ψ ζδy ψ
− δy ψ ζδx ψ − ψδy ζ δx ψ
y xy − δy ψ ζδx ψ (10.64)
xy x yx xy y − ζδy ψ δx ψ + ζδx ψ δy ψ(10.65) − ∆ x2
y
δx ψ δx ζ δy ψ
4 − ∆y 2
x
δy ψ δy ζ δx ψ
4 (10.66) Similarly, it can be shown that the average of J1 and J2 conserves enstrophy:
ζ J1 + J2
∆x2
x
yx
y
= δx ζ ζδy ψ −
δx ζ δx ζ δy ψ
2
4 y − δy ζ ζδx ψ xy − ∆y 2
x
δy ζ δy ζ δx ψ
4
(10.67) 160 CHAPTER 10. NONLINEAR EQUATIONS
Notice that the ﬁnite diﬀerence Jacobians satisfy the following property:
J1 (ψ, ζ ) = −J1 (ζ, ψ ) J2 (ψ, ζ ) = −J3 (ζ, ψ ). (10.68)
(10.69) Hence, from equation 10.66 ζJ3 (ζ, ψ ) can be written in ﬂux form, and from equation
10.67 ψ J1 +J3 can also be written in ﬂux form. These results can be tabulated:
2
energy conserving
J2
J1 + J3
2 enstrophy conserving
J3
J1 + J2
2 Notice that any linear combination of the energy conserving schemes will also be energy
conserving, likewise for the enstrophy conserving forms. Thus, it is possible to ﬁnd an
energy and enstrophy conserving Jacobian if we can ﬁnd two constants α and β such
that:
J1 + J2
J1 + J3
= βJ3 + (1 − β )
(10.70)
JA = αJ2 + (1 − α)
2
2
Equating like terms in J1 , J2 and J3 we can solve the system of equation. The ﬁnal
result can be written as:
J1 + J2 + J3
(10.71)
JA =
3
Equation 10.71 deﬁnes the Arakawa Jacobian, named in honor of Akio Arakawa who
proposed it ﬁrst. The expression for JA in terms of the FD computational stencil is a
little complicated. We give the expression for a square grid spacing:∆x = ∆y .
12∆x∆yJA (ζ, ψ ) = (ζj +1,k + ζj,k ) (ψj +1,k+1 + ψj,k+1 − ψj +1,k−1 − ψj,k−1 ) + (ζj,k+1 + ζj,k ) (ψj −1,k+1 + ψj −1,k − ψj +1,k+1 − ψj +1,k )
+ (ζj −1,k + ζj,k ) (ψj −1,k−1 + ψj,k−1 − ψj −1,k+1 − ψj,k−1 ) + (ζj,k−1 + ζj,k ) (ψj +1,k−1 + ψj +1,k − ψj +1,k−1 − ψj −1,k )
+ (ζj +1,k+1 + ζj,k ) (ψj,k+1 − ψj +1,k ) + (ζj −1,k+1 + ζj,k ) (ψj −1,k − ψj,k+1 ) + (ζj −1,k−1 + ζj,k ) (ψj,k−1 − ψj −1,k ) + (ζj +1,k−1 + ζj,k ) (ψj +1,k − ψj,k−1 ) (10.72) Note, the terms in ζj,k cancel out, the expression for JA can use ±ζj,k or no value at all.
An important property of the Arakawa Jacobian is that it inhibits the pile up of energy
at small scales, a consequence of conserving enstrophy and energy. Since both quantities
are conserved, so is their ratio which can be used to deﬁne an average wavenumber κ:
κ2 = ∇ψ 2 dA
.
2
2
A (∇ ψ ) dA
A (10.73) For the case of a periodic problem, the relationship between the above ratio and wavenumbers can be easily demonstrated by expanding the streamfunction and vorticity in terms 10.6. CONSERVATION IN PRIMITIVE EQUATIONS
cψ v
s us q ηj,k cψ v
s cψ 161 us
cψ Figure 10.4: Conﬁguration of unknowns on an Arakawa CGrid. The CGrid velocity
points ui+ 1 and vi,j + 1 are located a distance d/2 to the left and top, respectively, of
2
2
pressure point ηi,j .
of the Fourier components:
ˆ
ψm,n eimx einy ψ=
m ζ=− (10.74) n ˆ
(m2 + n2 )ψm,n eimx einy
m (10.75) n ˆ
where ψm,n are the complex Fourier coﬃcients and the computational domain has been
mapped into the square domain [0, 2π ]2 . Using the orthogonality of the Fourier modes,
it is easy to show that the ratio κ becomes
κ2 = m
m ˆ
+ n2 )2 ψm,n 2
2
2ˆ
2
n (m + n )ψm,n  n (m 2 (10.76) The implication of equation 10.76 is that there can be no oneway cascade of energy in
wavenumber space; if some “local” cascading takes place from one part of the spectrum
to the other; there must be a compensating shift of energy in another part. 10.6 Conservation in primitive equations The Arakawa Jacobian enforces enstrophy and energy conservation when the vorticitystreamfunction formulation is discretized. There are situatiions (in the presence of islands, for example) where the vorticitystreamfunction formulation is not appropriate
and one must revert to discretizing the primitive equations (momentum and volume
conservation). The question becomes what is the appropriate FD discretization of the
nonlinear momentum advection that can guarantee conservation of kinetic energy and
enstrophy? The process to obtaining these diﬀerencing schemes is to backtrap the steps
that lead to the derivation of the Jacobian operator in the vorticity equation. The
Jacobian in the vorticity equation arises from the crossdiﬀerentiation of the nonlinear
advection terms:
∂ v · ∇v ∂ v · ∇u
−
∂x
∂y =
= ∂
∂u
∂v
∂v
∂
∂v
−u
−v
u
+
v
∂x
∂x
∂y
∂y
∂x
∂y
∂v
∂u
∂
∂v
∂
∂u
+v
+v
u
−
u
∂x
∂x
∂y
∂y
∂x
∂y (10.77)
(10.78) 162 CHAPTER 10. NONLINEAR EQUATIONS In order to make use of the results obtained using the vorticitystreamfunction form,
it is usefull to introduce a ﬁctitious streamfunction in the primitive variables using a
staggered grid akin to the Arakawa Cgrid shown in ﬁgure 10.4. The staggered velocity
are deﬁned with respect to the “ﬁctitious streamfunction”:
u = −δy ψ, v = δx ψ, ζ = δx v − δy u (10.79) The energy conserving Jacobian can thus be written as:
x J2 (ζ, ψ ) = −δx uy δx v + v y δy v y x y + δy ux δx u + v x δy u (10.80) Comparing equations 10.80 and 10.77 we can deduce the following energy conserving
momentum advection operators:
x y v · ∇u = ux δx u + v x δy u
x
y
v · ∇v = uy δx v + v y δy v (10.81) In a similar manner, the enstrophy conserving Jacobian can be rewritten as:
J1 + J2
= −δx (uxy δx v x + v yy δy v y ) + δy (uxx δx ux + v xy δy uy ) ,
2 (10.82) and we can deduce the following enstrophyconserving operators:
v · ∇u = uxx δx ux + v xy δy uy
v · ∇v = uxy δx v x + v yy δy v y (10.83) If either 10.81 or 10.83 is used in the momentum equation, and if the ﬂow is discretely
divergencefree, then energy or enstrophy is conserved in the same manner as it is in the
vorticity equation through J2 or J1 +J2 . Stated diﬀerently, only the divergent part of the
2
velocity ﬁeld is capable of creating or destroying energy or enstrophy, in perfect analogy
to the behavior of the continuous equations.
We would like to have an operator that conserves both energy and enstrophy, which
means converting J3 . This is considerably harder. We skip the derivation and show the
result, see Arakawa and Lamb for details Arakawa and Lamb (1977)):
1
2
y′
y′
[δx (uxyy ux ) + δy (v xyy uy )] +
δx′ u′ ux + δy′ v ′ uy
3
3
2
1
x′
x′
xxy x
xxy y
∇ · vv =
[δx (u v ) + δy (v v )] +
δx′ u′ v x + δy′ v ′ v y
3
3
ux + v y
u′ = −δy′ ψ = √
2
− ux + v y
√
v ′ = δx′ ψ =
2 ∇ · vu = (10.84)
(10.85)
(10.86)
(10.87) The (x′ , y ′ ) coordinate system is rotated 45 degrees counterclockwise to the (x, y ) coordinate system; i.e. it is in the diagonal directions w.r.t. to the original axis. 10.7. CONSERVATION FOR DIVERGENT FLOWS 10.7 163 Conservation for divergent ﬂows So far we have dealt mostly with advection operators which handle conservation laws
appropriate for divergencefree ﬂows. There are situations, such as ﬂows over obstacles,
where the divergent velocity plays a signiﬁcant role. Under such conditions, it might
be important to incorporate conservation laws appropriate to divergent ﬂows. In the
following we will consider primarily the shallow water equations in a rotating frame:
∂v
+ v · ∇v + f k × v + g ∇η = 0
∂t
∂h
+ ∇ · (hv) = 0
∂t (10.88)
(10.89) where the ﬂuid depth h is the sum of the resting layer thickness H and the surface
displacement η . One of the more important conservation principles is the conservation
of potential vorticity, q :
ζ+f
∂q
+ v · ∇q = 0, q =
(10.90)
∂t
h
The latter is derived by rewriting the nonlinear advection of momentum in the form
v · ∇v = ∇ v·v
− v × ζk
2 (10.91) prior to taking the curl of the momentum equation 10.88 to arrive at:
∂ζ + f
+ ∇ · [(ζ + f )v] = 0.
∂t (10.92) The potential vorticity conservation equation is obtained after expanding equation 10.92
and using the continiuity equation 10.89. Equation 10.92 is the ﬂux form of equation
10.90, and shows that the area average of hq is conserved if the domain is closed.
The best conﬁguration to solve the shallow water equations is that of the Cgrid,
ﬁgure 10.4; the vorticity, streamfunction, and potential vorticity are collocated. In terms
of the discrete operators we have
ζ = δx v − δy u, q = ζ+f
η xy (10.93) The ﬁnite diﬀerence discretization of the continuity equation on the Cgrid takes the
form:
∂h
+ δx U + δy V = 0
(10.94)
∂t
x y where U = h u and V = h v .
The purpose is to formulate FD expressions for the momentum advection terms compatible with the conservation of PV, equation 10.90. With this in mind, we start by
averaging equation 10.94 with xy , to bring it to the q collocation points, and multiply
the resulting equation by q to obtain:
xy ∂q h
∂t + δx q x U xy + δy q y V xy xy =h x
y
1
∂q
xy
xy
+ xy U δx q + V δy q
∂t h (10.95) 164 CHAPTER 10. NONLINEAR EQUATIONS Equation 10.95 is a FD expression of the identity
∂qh
Dq
= ∇ · (qhv) = h
∂t
Dt (10.96) If the right hand side of 10.95 is a FD expression for the PV equation 10.90, the left hand
side is a FD analogue to the vorticity equation 10.92. Carrying the steps backwards we
have the following component forms for −v × (ζ + f )k:
Continuum
−v (ζ + f )
u(ζ + f ) Discrete
xy
−V q y
xy
U qx ·
The remaining task is to ﬁnd suitable forms for the ∇ v2v . The choices available are either squaring the spaceaveraged velocity components, or averaging the squared velocity.
The latter however leads to a straightforward FD analogue of the kinetic energy and is
therefore preferred. This leads to the following PVconserving momentum advection and
Coriolis force operators: v · ∇u − f v =
v · ∇v + f u = 1
x
y
δx u2 + v 2 − v xy q y
2
1
x
y
δy u2 + v 2 + uxy qx
2 (10.97)
(10.98) It can be shown that the above operator also conserves potential enstrophy hq 2 /2.
The derivation of schemes that conserve both PV and kinetic energy is very complex.
Arakawa and Lamb Arakawa and Lamb (1981); Arakawa and Hsu (1981) did derive such
a diﬀerencing scheme. Here we quote the ﬁnal result:
x
1
1
1
1
x
y
xx
y
x
(10.99)
δx u2 + v 2 − V q xy − δ′ x (δ′ y V )δ′ x δ′ y q + δ′ x U δ′ y q x + δ′ x U δ′ y q x
2
48
12
12
y
1′
1
1
1′
xy
y′ y
′
′′
2 x + v 2 y − U x q xy +
(10.100)
δy u
δ y (δ x U )δ x δ y q − δ y V δ x q − δ′ y V δ′ x q y
2
48
12
12 where δ′ is the discrete diﬀerential operator without division by grid distance. Chapter 11 Special Advection Schemes
11.1 Introduction This chapter deals with specialized advection schemes designed to handle problems where
in addition to consistency, stability and conservation, additional constraints on the solution must be satisﬁed. For example, biological or chemical concentration must be
nonnegative for phsical reason; however, numerical errors are capable of generating negative values which are simply wrong and not amenable to physical interpretation. These
negative values can impact the solution adversely, particularly if there is a feed back
loop that exacerbate these spurious values by increasing their unphysical magnitude. An
example of a feed back loop is a reaction term valid for only positive values leading to
moderate growth or decay of the quantity in question; whereas negative values lead to
unstable exponential growth. Another example is the equation of state in ocean models
which intimately ties salt, temperature, and density. This equation is empirical in nature
and is valid for speciﬁc ranges of temperature, salt, and density; and the results of out
of range inputs to this equation are unpredictable and lead quickly to instabilities in the
simulation.
The primary culprit in these numerical artifacts is the advection operator as it is the
primary means by which tracers are moved around in a ﬂuid environment. Molecular
diﬀusion is usually too weak to account for much of the transport, and what passes for
turbulent diﬀusion has its roots in “vigorous” advection in straining ﬂow ﬁelds. Advection
transports a tracer from one place to another without change of shape, and as such
preserves the original extrema (maxima and minima) of the ﬁeld for long times (in the
absence of other physical mechanism). Problems occur when the gradient are too steep to
be resolved by the underlying computational grid. Examples include true discontinuities,
such as shock waves or tidal bore, or pseudidiscontinuities such as narrow temperature
or salt fronts that are too narrow to be resolved on the grid, (a few hundered meters
whereas the computational grid is of the order of kilometers).
A number of special advection schemes were devised to address some or all of these
issues. They are known generically as Total Variation Diminishing (TVD) schemes. They
occupy a prominent place in the study and numerical solution of hyperbolic equations
like the Euler equations of gas dynamics or the shallow water equations. Here we conﬁne
ourselves to the pure advection equation, a scalar hyperbolic equation.
165 166 11.2 CHAPTER 11. SPECIAL ADVECTION SCHEMES Monotone Schemes The properties of the pure advection operator to preserve the original extrema of the
advected ﬁeld is referred to as the monotonicity property. Consider an initially discretized
initial condition of the form Tj0 ≥ Tj0+1 , then a scheme is called monotone if
Tjn ≥ Tjn
+1 (11.1) for all j and n. A general advection scheme can be written in the form:
q Tjn+1 αk Tjn k
+ = (11.2) k =−p where the αk are coeﬃcients that depend on the speciﬁc scheme used. A linear scheme
is one where the coeﬃcients αk are independent of the solution Tj . For a scheme to be
n
monotone with respect to the Tk , we need the condition
∂Tjn+1
≥0
∂Tjn k
+ (11.3) Godunov has shown that the only linear monotonic scheme is the ﬁrst order (upstream)
donor cell scheme. All highorder linear schemes are not monotonic and will permit
spurious extrema to be generated. Highorder schemes must be nonlinear in order to
preserve monotonicity. 11.3 Flux Corrected Transport (FCT) The FCT algorithm was originally proposed by Boris and Book Boris and Book (1973,
1975, 1976) and later modiﬁed and generalized to multidimensions by Zalesack Zalesak
(1979). Here we present the Zalesak version as it is the more common one and ﬂexible one.
We will ﬁrst consider the scheme in onedimension before we consider its twodimensional
extension. 11.3.1 OneDimensional Consider the advection of a tracer in onedimension written in conservation form:
Tt + (uT )x = 0 (11.4) subject to appropriate initial and boundary conditions. The spatially integrated form of
this equation lead to the following:
xj + 1
2 xj − 1
2 where f x 1
j+ 2 Tt dx + f x j+ 1
2 − f x j− 1
2 =0 (11.5) = [uT ]xj− 1 is the ﬂux out of the cell j . This equation is nothing but the
2 restatement of the partial diﬀerential equation as the rate at which the budget of T in 11.3. FLUX CORRECTED TRANSPORT (FCT) 167 cell j increases according to the advective ﬂuxes in and out of the cell. As a matter of
fact the above equation can be reintrepeted as a ﬁnite volume method if the integral is
∂T
replaced by ∂tj ∆x where T j refers to the average of T in cell j whose size is ∆x. We
now have:
fj + 1 − fj − 1
∂T j
2
2
+
=0
(11.6)
∂t
∆x
If the analytical ﬂux is now replaced by a numerical ﬂux, F , we can generate a family
of discrete schemes. If we choose an upstream biased scheme where the value within each
cell is considered constant, i.e. Fj + 1 = uj + 1 Tj for uj + 1 > 0 and Fj + 1 = uj + 1 Tj +1 for
2
2
2
2
2
uj + 1 < 0 we get the donor cell scheme. Note that the two cases above can be rewritten
2
(and programmed) as:
Fj + 1 =
2 uj + 1 + uj + 1 
2 2 2 Tj + uj + 1 − uj + 1 
2 2 2 Tj +1 (11.7) The scheme will be monotone if we were to advance in time stably using a forward Euler
method. If on the other hand we choose to approximate Tj at the cell edge as the average
of the two cells:
Tj + Tj +1
Fj + 1 = uj + 1
(11.8)
2
2
2
we obtained the second order centered in space scheme. Presumably the second order
scheme will provide a more accurate solution in those regions where the advected proﬁle
is smooth whereas it will create spurious oscillations in regions where the solution is
“rough”.
The idea behind the ﬂux corrected transport algorithm is to use a combination of
the higher order ﬂux and the lower order ﬂux to prevent the generation of new extrema.
The algorithm can be summarized as follows:
1. compute low order ﬂuxes FjL 1 .
+
2 2. compute high order ﬂuxes FjH 1 , e.g. second order interpolation of T to cell edges
+
2 or higher. 3. Deﬁne the antidiﬀusive ﬂux Aj + 1 = FjH 1 − FjL 1 . This ﬂux is dubbed antidiﬀuse
+
+
2 2 2 because the higher order ﬂuxes attempt to correct the over diﬀusive eﬀects of the
low order ﬂuxes. 4. Update the solution using the low order ﬂuxes to obtain a ﬁrst order diﬀused but
monotonic approximation:
Tjd = Tjn − FjL 1 − FjL 1
+
−
2 2 ∆x ∆t (11.9) 5. Limit the antidiﬀusive ﬂux so that the corrected solution will be free of extrema not
found in Tjn or Tjd . The limiting is eﬀected through a factor: Ac+ 1 = Cj + 1 Aj + 1 ,
j
where 0 ≤ Cj + 1 ≤ 1.
2 2 2 2 168 CHAPTER 11. SPECIAL ADVECTION SCHEMES 6. Apply the antidiﬀusive ﬂux to get the corrected solution
Tjn+1 = Tjd − Ac+ 1 − Ac− 1
j
j
2 2 ∆x ∆t (11.10) Notice that for C = 0 the antidiﬀusive ﬂuxes are not applied, and we end up with
Tjn+1 = Tjd ; while for C = 1, they are applied at full strength. 11.3.2 OneDimensional Flux Correction Limiter In order to elucidate the role of the limiter we expand the last expression in terms of the
high and low order ﬂuxes to obtain: Tjn+1 = Tjn − Cj + 1 FjH 1 + 1 − Cj + 1 FjL 1 − Cj − 1 FjH 1 + 1 − Cj − 1 FjL 1
+
−
+
−
2 2 2 2 2 2 2 2 ∆x ∆t (11.11)
The term under the bracket can thus be interpreted as a weighed average of the low and
high order ﬂux; and the weights depend on the local smoothness of the solution. Thus for
a rough neighborhood we should choose C → 0 to avoid oscillations, while for a smooth
neighborhood C = 1 to improve accuracy. As one can imagine the power and versatility
of FCT lies in the algorithm that prescribe the limiter. Here we prescribe the Zalesak
limiter.
1. Optional step designed to eliminate correction near extrema. Set Aj + 1 = 0 if:
2 Aj + 1 Tjd+1 − Tjd < 0 and
2 A 1 Tjd+2 − Tjd+1 < 0 j+ 2 or A 1 Td − Td
j
j −1 < 0
j+ (11.12) 2 2. Evaluate the range of permissible values for Tjn+1 : T max = max T n , T n , T n , T d , T d , T d
j
j −1 j
j +1 j −1 j
j +1 T min = min T n , T n , T n , T d , T d , T d
j +1
j +1 j −1 j
j −1 j
j (11.13) 3. Compute the antidiﬀusive ﬂuxes Pj+ going into cell j : Pj+ = max 0, Aj − 1 − min 0, Aj + 1
2 2 (11.14) These incoming ﬂuxes will increase Tjn+1 .
4. Compute the maximum permissible incoming ﬂux that will keep Tjn+1 ≤ Tjmax .
From the corrective step in equation 11.10 this given by
Q+ = Tjmax − Tjd
j ∆x
∆t (11.15) 11.3. FLUX CORRECTED TRANSPORT (FCT) 169 5. Compute limiter required so that the extrema in cell j are respected:
+
Rj + min 1, Qj
=
Pj+ 0 if Pj+ > 0 if Pj+ = 0 (11.16) 6. Steps 3, 4 and 5 must be repeated so as to ensure that the lower bound on the
solution Tjmin ≤ Tjn+1 . So now we deﬁne the antidiﬀusive ﬂuxes away from cell j :
Pj− = max 0, Aj + 1 − min 0, Aj − 1
2 2 Q− = Tjd − Tjmin
j −
Rj − min 1, Qj
=
Pj− 0 ∆x
∆t (11.17)
(11.18) if Pj− > 0 if Pj− (11.19) =0 7. We now choose the limiting factors so as enforce the extrema constraints simultaneously on adjacent cells.
Cj + 1 =
2 11.3.3 Properties of FCT min R+ , R−
j +1
j min R+ , R−
j
j +1 if Aj + 1 > 0
2 if Aj + 1 < 0 (11.20) 2 Figure 11.1 shows the advection of a function using a centered diﬀerencing formula of
order 8 using 50, 100, 200, 400 and 2000 points around a periodic domain of length 20.
The red and blue curves show the analytical and the numerical solution, respectively.
The timestepping is based on an RK3 scheme with a ﬁxed time step of ∆t = 10−3 .
The function is composed of multiple shapes to illustrate the strength and weaknesses of
diﬀerent numerical discretization on various solution proﬁles with diﬀerent levels of discontinuities: square and triangular waves, a truncated inverted parabola with a smooth
maximum, and an inﬁnitely smooth narrow Gaussian. The discontinuities in the proﬁle
are challenging for highorder method and this is reﬂected in the solutions obtained in
11.1 where Gibbs oscillations pollute the entire solution and at all resolutions. One can
anticipate that in the limit of inﬁnite resolution the amplitude of these Gibbs oscillations
will reach a ﬁnite limit independent of the grid size. The top solution with 50 points
is severely underresolved (even the analytical solution does not look smooth since it is
drawn on the numerical grid). The situation improves dramatically with increased resolution for the smooth proﬁles where the smooth peaks are now wellrepresented using
∆x = 20/200 = 0.1. The Gibbs oscillations seen riding on the smooth peaks are caused
by the spurious dispersion of grid scale noise. The resolution increase does not pay oﬀ
for the square wave where the numerical solution exhibit noise at all resolutions.
In contrast to the highorder solution, ﬁgure 11.2 shows the results of the same
computations using a donorcell scheme to compute the ﬂuxes. As anticipated from our 170 CHAPTER 11. SPECIAL ADVECTION SCHEMES 1.2 M=50 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=100 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=200 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=400 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=2000 1 0.8 0.6 0.4 0.2 0 −0.2
−10 −8 −6 −4 −2 0 2 4 6 8 Figure 11.1: Uncorrected Centered Diﬀerence of 8th order with RK3, ∆t = 10−3 . 10 11.3. FLUX CORRECTED TRANSPORT (FCT) 171 1.2 M=50 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=100 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=200 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=400 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=2000 1 0.8 0.6 0.4 0.2 0 −0.2
−10 −8 −6 −4 −2 0 2 4 6 8 Figure 11.2: upwind 1st order scheme (donorcell) with RK1, ∆t = 10−3 . 10 172 CHAPTER 11. SPECIAL ADVECTION SCHEMES 1.2 M=50 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=100 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=200 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=400 1 0.8 0.6 0.4 0.2 0 −0.2
1.2 M=2000 1 0.8 0.6 0.4 0.2 0 −0.2
−10 −8 −6 −4 −2 0 2 4 6 8 Figure 11.3: FCTCentered Diﬀerence scheme of 8th order with RK3, ∆t = 10−3 . 10 11.3. FLUX CORRECTED TRANSPORT (FCT) 173 theoretical discussion of the donorcell scheme the errors are primarily dissipative. At
coarse resolution the signals in the solution have almost disappeared and the solution
has been smeared excessively. Even at the highest resolution used here the smooth peaks
have lost much of their amplitudes while the discontinuities (of various strengthes) have
also been smeared. On the upside the donorcell scheme has delivered solutions that are
oscillationsfree and that respect the TVD properties of the continuous PDE.
The FCT approach, as well as other limiting methods, aim to achieve a happy medium
between diﬀusive but oscillationfree and highorder but oscillatory. Figure 11.3 illustrate FCT’s beneﬁts (and drawbacks). First the oscillations have been eliminate at all
resolutions. Second, at the coarseest resolution, there is a signiﬁcant loss of peak accompanied by the socalled terracing eﬀect where the limiter tends to ﬂatten smooth peaks,
and to introduce intermediate ”step” in the solution where previously there was none.
This eﬀect continues wellinto the well resolved regime of ∆x = 20/400 where the smooth
Gaussian peak has still not recovered its full amplitude and is experiencing a continuiing
terracing eﬀect. This contrasts with the uncorrected solution where the smooth peak
was recoved with a more modest resolution of ∆x = 20/200. This is a testament to an
overactive FCT that cannot distinguish between smooth peaks and discontinuities. As
a result smooth peaks that ought not to be limited are ﬂattened out. It is possible to
mitigate an overzealous limiter by appending a “discriminator” that can turn oﬀ the limiting near smooth extrema. In general the discriminator is built upon checking whether
the second derivative of the solution is onesigned in neighborhoods where the solution’s
slope changes sign. Furthermore, the terracing eﬀect can be eliminated by introducing a
small amount of scale selective dissipation ﬂux (essentially a hyperviscosity). For further
details check out Shchepetkin and McWilliams (1998) and Zalesak (2005).
The FCT procedure has advantages and disadvantages. The primary beneﬁt of the
procedure is that it is a practical procedure to prevent the generation of spurious extrema. It is also ﬂexible in deﬁning the high order ﬂuxes and the extrema of the ﬁelds.
Most importantly the algorithm can be extended to multiple dimensions in a relatively
straightforward manners. The disadvantages is that the procedure is costly in CPU
compared to unlimited method, but hey nothing comes free. 11.3.4 TwoDimensional FCT In two dimensions the advection equation takes the form Tt + fx + gy = 0, where f = uT
and g = vT . The FCT algorithm takes the same form as before after taking account of
the extra ﬂux G and extra spatial dimension. We thus have the low order solution:
d
Tj,k n
Tj,k FjL 1 ,k − FjL 1 ,k
−
+ (11.21) Aj + 1 ,k = FjH 1 ,k − FjL 1 ,k , Aj,k+ 1 = GH + 1 − GL + 1
+
j,k
+
j,k (11.22) − ∆t 2 GL + 1 − GL − 1
j,k
j,k
2 = 2 ∆x − ∆t 2 ∆y and the following antidiﬀusive ﬂuxes:
2 2 2 2 2 2 max
The extrema of the solution are deﬁned in two passes over the data. First we set Tj,k =
n
d
max(Tj,k , Tj,k ). The ﬁnal permissible values are determined by computing the extrema 174 CHAPTER 11. SPECIAL ADVECTION SCHEMES of the previous ﬁelds in the neighboring cells: T max = max T max , T max , T max
j,k
j,k
j ±1,k j,k ±1 (11.23) T min = min T min , T min , T min
j,k
j,k
j ±1,k j,k ±1 Finally, the incoming and outgoing ﬂuxes in cell (j, k) are given by:
+
Pj,k = max 0, Aj − 1 ,k − min 0, Aj + 1 ,k + max 0, Aj,k− 1 − min 0, Aj,k+ 1 (11.24)
2 −
Pj,k 2 2 2 = max 0, Aj + 1 ,k − min 0, Aj − 1 ,k + max 0, Aj,k+ 1 − min 0, Aj,k− 1 (11.25) 11.3.5 2 2 2 2 TimeDiﬀerencing with FCT A last practical issue with the implementation of FCT is the choice of timediﬀerencing
for the combined low and high order scheme. In order to preserve the monotone property
the low order scheme must rely on a ﬁrst order Forward Euler scheme in time. For the
high order ﬂux it is desirable to increase the order of the timediﬀerencing to match that
of the spatial diﬀerencing, and to increase the timeaccuracy at least in smooth regions.
There is an additional wrinkle to this dilemna in that whereas Forward Euler in time
is stable for the donorcell scheme, it is unconditionally unstable for centered in space
spatial diﬀerences.
The resolution of this dilemna can be addressed in several ways. One approach is
to use a RungeKutta type approach and to use the low and high order ﬂuxes at each
of the substages of the Runge Kutta scheme. For the RK scheme to be stable to both
spatial diﬀerences; we need at least a third order scheme (their stability region includes
portions of the imaginary axis and the left hand complex planes). The drawback of such
an approach is an increase in CPU time since multiple evaluations of the right hand sides
is required. The CPU cost is exacerbated if the FCT limiter is applied at each of the
substeps.
Another approach consists of using a multilevel methods like the leapfrog trapezoidal
scheme. The low ﬂux is ﬁrst updated using the solution at time n. The high order ﬂuxes
at time n are obtained with a traditional leapfrog trapezoidal step that does not involve
the low order solution (i.e. no limiter is applied): ﬁrst the leapfrog step is applied:
ˆ
T n+1 = T n−1 − ∆t FjH 1 (T n ) − FjH 1 (T n )
−
+
2 2 ∆x (11.26) A second order estimate of the function values at the midtime level is calculated:
1 T n+ 2 = ˆ
T n + T n+1
2 (11.27) and then used to compute the high order ﬂuxes
1 FjH 1 = FjH 1 T n+ 2 .
+
+
2 2 It is these last high order ﬂuxes that are limited using the FCT algorithm. (11.28) 11.4. SLOPE/FLUX LIMITER METHODS 175 3
2.5
MC 2
C(r) e
rbe
upe
S
Van Leer
minmod 1.5
1
0.5
0
0 1 r 2 3 Figure 11.4: Graph of the diﬀerent limiters as a function of the slope ratio r . 11.4 Slope/Flux Limiter Methods The slope/ﬂux limiter class of methods uses a similar approach to the ﬂux correction
method, in that a low order and a high order ﬂux is used to eliminate spurious oscillations.
The slopelimiter schemes however do not involve the computations of the temporary
diﬀused value. The limiting is instead applied directly to the ﬂux based on the values of
the solution’s gradients. That the ﬁnal ﬂux used in the scheme takes the form:
Fj + 1 = FjL 1 + Cj + 1 FjH 1 − FjL 1
+
+
+
2 2 2 2 (11.29) 2 where the limiting factor C = C (r ) is a function of the slope ratio in neighboring cells:
rj + 1 =
2 Tj − Tj −1
.
Tj +1 − Tj (11.30) For slowly varying smooth function the slope ratio is close to 1; the slopes change sign
near an extremum and the ratio is negative. A family of method can be generated based
entirely on the choice of limiting function; here we list a few of the possible choices:
MinMod:
Superbee:
Van Leer:
MC: C (r ) = max(0, min(1, r ))
C (r ) = max(0, min(1, 2r ), min(2, r ))
r+
C (r ) = 1+r
r 
C (r ) = max(0, min(2r, 1+2r , 2)
2 (11.31) The graph of these limiters is shown in ﬁgure 11.4. The diﬀerent functions have a
number of common features. All limiters set C = 0 near extrema (r ≤ 0). They all
asymptote to 2, save for the minmod limiter which asymptotes to 1, when the function
changes rapidly (r ≫ 1). The minmod limiter is the most stringent of the limiters and
prevents the solution gradient from changing quickly in neighboring cells; this limiter is
known as being diﬀusive. The other limiters are more lenient, the MC one being the 176 CHAPTER 11. SPECIAL ADVECTION SCHEMES most lenient, and permit the gradients in neighboring cells to be twice as large as the
one in the neighboring cell. The Van Leer limiter is the smoothest of the limiters and
asymptotes to 2 for r → ∞. 11.5 MPDATA The Multidimensional Positive Deﬁnite Advection Transport Algorithm (MPDATA) was
presented by Smolarkiewicz (1983) as an algorithm to preserve the positivity of the ﬁeld
throughout the simulation. The motivation behind his work is that chemical tracers
must remain positive. Nonoscillatory schemes like FCT are positive deﬁnite but are
deemed too expensive, particularly since oscillations are tolerable as long as they did not
involve negative values. MPDATA is built on the monotone donor cell scheme and on
its modiﬁed equation. The latter is used to determine the diﬀusive errors in the scheme
and to correct for it near the zero values of the ﬁeld. The scheme is presented here in
its onedimensional form for simplicity. The modiﬁed equation for the donor cell scheme
where the ﬂuxes are deﬁned as in equaiton 11.7 is:
∂uT
∂
∂T
∂T
+
=
κ
∂t
∂x
∂x
∂x + O(∆x2 ) (11.32) where κ is the numerical diﬀusion generated by the donor cell scheme:
κ= u∆x − u2 ∆t
2 (11.33) The donor cell scheme will produce a ﬁrst etimate of the ﬁeld which is guranteed
to be nonnegative if the initial ﬁeld is initially nonnegative. This estimate however, is
too diﬀused; and must be corrected to eliminate these ﬁrst order errors. MPDATA data
achieves the correction by casting the second order derivatives in the modiﬁed equation
11.32 as another transport step with a pseudovelocity u:
˜ κ ∂T
∂ uT
˜
∂T
=−
, u=
˜ T ∂x
∂t
∂x
0 T >0 (11.34) T =0 and reusing the donor cell scheme to discretize it. The velocity u plays the role of an
˜
antidiﬀusion velocity that tries to compensate for the diﬀusive error in the ﬁrst step.
The correction step takes the form:
uj + 1
˜ = ˜
Fj + 1 = 2 2 uj + 1 ∆x − u2+ 1 ∆t
j
2 2 uj + 1 + uj + 1
˜
˜
2 ˜
Tjn+1 = Tj − 2 Tjd + 2
˜
˜
Fj + 1 − Fj − 1
2 2 ∆x Tjd+1 − Tjd (Tjd+1 + Tjd + ǫ)∆x uj + 1 − uj + 1
˜
˜ ∆t 2 2 2 Tjd+1 (11.35) (11.36)
(11.37) where Tjd is the diﬀused solution from the donorcell step, and ǫ is a small positive
number, e.g. 10− 15, meant to prevent the denominator from vanishing when Tjd = 11.6. WENO SCHEMES IN VERTICAL 177 Tjd+1 = 0. The second donor cell step is stable provided the original one is too; and
hence the correction does not penalize the stability of the scheme. The procedure to
derive the twodimensional version of the scheme is similar; the major diﬃculty is in
deriving the modiﬁed equation and the corresponding antidiﬀusion velocity. It turns
out that the xcomponent of the antidiﬀusion velocity remains the same while the ycomponents takes a similar form with u replaced by v , and ∆x by ∆y . 11.6 WENO schemes in vertical We explore the application of WENO methodology to compute in the vertical. The
WENO methodology is based on a reconstruction of function values from cells averages
using diﬀerent stencils, and on combining the diﬀerent estimates so as to maximime accuracy while minimizing the impact of nonsmooth stencils. We brieﬂy describe the steps
of a WENO calculation below, the details can be found in Shu (1998); Jiang and Shu
(1996). Note that the diﬀusive term requires the calculation of the derivative of the function. This can be also done accurately with a WENO scheme after the reconstruction of
T ; the caveat for high order accuracy is that the grid spacing ∆z must vary smoothly.
We ﬁrst take up the reconstruction step and dwell on diﬀerentiation later. The question
is of course always how much should we pay for an accurate calculation of the vertical
diﬀusion term. 11.6.1 Function reconstruction
s i−l s ··· s i−1 zi− 1 s i 2 ' ∆zi zi+ 1
2 E s i+1 s ··· s i+s Figure 11.5: Sketch of the stencil S (i; k, l). This stencil is associated with cell i, has left
shift l, and contains k = l + s + 1 cells.
We ﬁrst focus on the issue of computing function values from cell averages. We divide
the vertical into a number of ﬁnite volumes which we also refer to as cells, and we deﬁne
cells, cell centers and cells sizes by:
(11.38) Ii = zi− 1 , zi+ 1
2 zi = 2 zi− 1 + zi+ 1
2 ∆zi = zi+ 1
2 2 2
− zi− 1 2 (11.39)
(11.40) The reconstruction problem can be stated as follows: Given the cell averages of a
function T (z ):
zi+ 1
1
z
2
v (z ′ ) dz ′ , i = 1, 2, . . . , N
(11.41)
Ti =
∆zi zi− 1
2 178 CHAPTER 11. SPECIAL ADVECTION SCHEMES ﬁnd a polynomial pi (z ), of degree at most k − 1, for each cell i, such that it is a kth
order accurate approximation to the function T (z ) inside Ii :
pi (z ) = T (z ) + O(∆z k ), z ∈ Ii , i = 1, 2, . . . , N (11.42) The polynomial pi (z ) interpolates the function within cells. It also provides for a discontinuous interpolation at cell boundaries since a cell boundary is shared by more then
one cell; we thus write:
Ti+ 1 = pi (zi− 1 ), Ti− 1 = pi (zi+ 1 )
−
+
2 2 (11.43) 2 2 Given the cell Ii and the order of accuracy k, we ﬁrst choose a stencil, S (i; k, l), based
on Ii , l cells to the left of Ii and s cells to the right of Ii with l + s + 1 = k. S (i) consists
of the cells:
S (i) = {Ii−l , Ii−l+1 , . . . , Ii+s }
(11.44)
There is a unique polynomial p(z ) of degree k − 1 = l + s, whose cell average in each of
the cells in S (i) agrees with that of T (z ):
z Tj = 1
∆zj zj + 1
2 zj − 1 p(z ′ ) dz ′ , j = i − l, . . . , i + s. (11.45) 2 The polynomial in question is nothing but the derivative of the Lagrangian interpolant
of the function T (z ) at the cell boundaries. To see this, we look at the primitive function
of T (z ):
z T (z ) = −∞ T (z ′ ) dz ′ , (11.46) where the choice of lower integration limit is immaterial. The function T (z ) at cell edges
can be expressed in terms of the cell averages:
i T (zi+ 1 ) =
2 zj + 1
2 j =−∞ zj − 1
2 i T (z ′ ) dz ′ = z T j ∆zj (11.47) j =−∞ Thus, the cell averages deﬁne the primitive function at the cell boundaries. If we denote
the unique polynomial of degree at most k which interpolates T at the k + 1 points:
zi−l− 1 , . . . , zi+s+ 1 , by P (z ), and denote its derivative by p(z ), it is easy to verify that:
2 2 1
∆zj zj + 1 2 zj − 1 p(z ′ ) dz ′ = 2 =
=
= 1
∆zj zj + 1
2 zj − 1 P ′ (z ′ ) dz ′ 2 P (zj + 1 ) − P (zj − 1 )
2 2 ∆zj
T (zj + 1 ) − T (zj − 1 )
2 2 ∆zj 1
∆zj
z (11.48) zj + 1
2 zj − 1 T (z ′ ) dz ′ (11.49)
(11.50)
(11.51) 2 = T j , j = i − l, . . . , i + s (11.52) 11.6. WENO SCHEMES IN VERTICAL 179 This implies that p(z ) is the desired polynomial. Standard approximation theory tells
us that P ′ (z ) = T ′ (z ) + O(∆z k ), z ∈ Ii , which is the accuracy requirement.
The construction of the polynomial p(z ) is now straightforward. We can start with
the Lagrange intepolants on the k + 1 cell boundary and diﬀerentiate with respect to z
to obtain: k p(z ) = m−1 m=0 j =0 z T i−l+j ∆zi−l+j k k n=0
q =0
n=m q =m,n z − zi−l+q− 1 2 k
n=0
n=m zi−l+m− 1 − zi−l+n− 1
2 2 (11.53) The order of the outer sums can be exchanged to obtain an alternative form which maybe
computationally more practical:
p(z ) = k −1
j =0 z Clj (z )T i−l+j (11.54) where Clj (z ) is given by: k Clj (z ) = ∆zi−l+j m=j +1 k k n=0
q =0
n=m q =m,n z − zi−l+q− 1
2 k
n=0
n=m zi−l+m− 1 − zi−l+n− 1
2 2 (11.55) The coeﬃcient Clj need not be computed at each time step if the computational grid is
ﬁxed, instead they can be precomputed and stored to save CPU time. The expression
for the Clj simpliﬁes (because many terms vanish) when the point z coincide with a cell
edge and/or when the grid is equally spaced (∆zj = ∆z, ∀j ).
ENO reconstruction
The accuracy estimate holds only if the function is smooth inside the entire stencil
S (i; k, l) used in the interpolation. If the function is not smooth Gibbs oscillations
appear. The idea behind ENO reconstruction is to vary the stencil S (i; k, l), by changing
the left shift l, so as to choose a discontinuityfree stencil; this choice of S (i; k, l) is called
an “adaptive stencil”. A smoothness criterion is needed to choose the smoothest stencil,
and ENO uses Newton divided diﬀerences. The stencil with the smoothest Newton
divided diﬀerence is chosen.
ENO properties:
1. The accuracy condition is valid for any cell which does not contain a discontinuity.
2. Pi (z ) is monotone in any cell Ii which does contain a discontinuity. 180 CHAPTER 11. SPECIAL ADVECTION SCHEMES 3. The reconstruction is Total Variation Bounded (TVB as opposed to TVD), that
k
is there is a function Q(z ) satisfying Q(z ) = Pi (z ) + O(∆zi +1 ), z ∈ Ii , such that
T V (Q) ≤ T V (T ).
ENO disadvantages:
1. The choice of stencil is sensitive to roundoﬀ errors near the roots of the solution
and its derivatives.
2. The numerical ﬂux is not smooth as the stencil pattern may change at neighboring
points.
3. In the stencil choosing process k stencils are considered covering 2k − 1 cells but
only one of the stencils is used. If information from all cells are used one can
potentially get 2k − 1th order accuracy in smooth regions.
4. ENO stencil choosing is not computationally eﬃcient because of the repeated use
of “if” structures in the code. 11.6.2 WENO reconstruction WENO attempts to address the disadvantages of ENO, primarily a more eﬃcient use of
CPU time to gain accuracy in smooth region without sacriﬁcing the TVB property in
the presence of discontinuity. The basic idea is to use a convex combination of all stencils
used in ENO to form a better estimate of the function value. Suppose the k candidate
stencils S (i; k, l), l = 0, . . . , k − 1 produce the k diﬀerent estimates:
Tjl+ 1 =
2 k −1
j =0 z Clj T i−l+j , l = 0, . . . , k − 1 then the WENO estimate is
Tj + 1 =
2 k −1 ωl Tjl+ 1 . l=0 (11.56) (11.57) 2 where ωl are the weights satisfying the following requirements for consistency and stability:
ωl ≥ 0, k −1 ωl = 1 (11.58) l=0 Furthermore, when the solution has a discontinuity in one or more of the stencils we
would like the corresponding weights to be essentially 0 to emulate the ENO idea. The
weights should also be smooth functions of the cell averages. The weights described
below are in fact C ∞ .
Shu et al propose the following forms for the weights:
ωl = αl
k −1
n=0 αn , αl = dl
l = 0, . . . , k − 1
(ǫ + βl )2 (11.59) 11.6. WENO SCHEMES IN VERTICAL 181 Here, the dl are the factor needed to maximize the accuracy of the estimate, i.e. to
make the truncation error O(∆z 2k−1 ). Note that the weights must be as close to dl in
smooth regions, actually we have the requirement that ωl = dl + O(∆z k ). The factor ǫ
is introduced to avoid division by zero, a value of ǫ = 10−6 seems standard. Finally, βl
are the smoothness indicators of stencils S (i; k, l). These factors are responsible for the
success of WENO; they also account for much of the CPU cost increase over traditional
methods. The requirements for the smoothness indicator are that βl = O(∆z 2 ) in smooth
regions and O(1) in the presence of discontinuities. This translates into αl = O(1) in
smooth regions and O(∆z 4 ) in the presence of discontinuities. The smoothness measures
advocated by Shu et al look like weighed H k−1 norms of the interpolating functions:
βl = k −1 zi+ 1 n=1 zi− 1
2 2 ∆z 2n−1 ∂ n pl
∂z n 2 dz (11.60) The right hand side is just the squares of the scaled L2 norms for all derivatives of the
polynomial pl over the interval [zi− 1 , zi+ 1 ]. The factor ∆z 2n−1 is introduced to remove
2
2
any ∆z dependence in the derivatives in order to preserve self similarity; the smoothness
indicator are the same regardless of the underlying grid. The smoothness indicators for
the case k = 2 are:
z
z
z
z
(11.61)
β0 = (T i+1 − T i )2 , β1 = (T i − T i−1 )2
Higher order formulae can be found in Shu (1998); Balsara and Shu (2000). The formulae
given here have a onepoint upwind bias in the optimal linear stencil suitable for a
problem with wind blowing from left to right. If the wind blows the other way, the
procedure should be modiﬁed symemetrically with respect to zi+ 1 .
2 11.6.3 ENO and WENO numerical experiments Figure 11.6 (left panel) shows the convergence rates of ENO interpolation for the function
1
sin 2πx for − 1 ≤ x ≤ 2 . Two sets of experiments were conducted. One set the shift to 0
2
so the interpolation is right tilted, and the other to (k − 1)/2 where k is the polynomial
order so that the stencil is centered. The two sets of experiments overlap for k = 2, 3.
The convergence rates for both experiments are the same, although the centered stencils
yield a lower error. The WENO reconstruction eﬀectively doubles the convergence rates
by using a convex combination of all stencils used in the reconstruction.
I have coded up a WENO advection scheme that can use variable order space interpolation (up to order 9), and up to 3rd order RungeKutta stepping. I have also
experimented with the scheme in 1D. Figure 11.7 shows the advection of Shchepetkin’s
narrow proﬁle (top left), wide proﬁle (top right), and hat proﬁle, (bottom left). The high
order WENO5RK3 scheme has less dissipation, and better phase properties than the
WENO3RK2 scheme. For the narrow Gaussian hill the peak is well preserved and the
proﬁle is narrower; it is indistinguishable from the analytical solution for the wider proﬁle. Finally, although the scheme does not enforce TVD there is no evidence of dispersive
ripples in the case of the hat proﬁle; there are however small negative values.
I have tried to implement the shape preserving WENO scheme proposed by Suresh and Huynh
(1997) and Balsara and Shu (2000). Their limiting attempts to preserve the high order 182
10 CHAPTER 11. SPECIAL ADVECTION SCHEMES
0 10 10 2
10 0 −5 −5 3
ε ε 10 3 −10 4
10 −10 5 10 −15 5 7 9 6 10 7 −15 10 1 2 10
N 10
10 3 −20 10 1 10 2 10 3 10 4 N Figure 11.6: Convergence Rate (in the maximum norm) for ENO (left panel) and WENO
(right panel) reconstuction. The dashed curves are for a left shift set to 0, while the solid
curve are for centered interpolation. The numbers refer to the interpolation order accuracy of WENO near discontinuities and smoth extrema, and as such include a peak
discriminator that picks out smooth extrema from discontinuous ones. As such, I think
the scheme will fail to preserve the peaks of the original shape and will allow some new
extrema to be generated. This is because there is no full proof discrimator. Consider
what happens to a square wave advected by a uniform velocity ﬁeld. The discontinuity
is initially conﬁned to 1 cell; the discriminator will rightly ﬂag it as a discontinuous extremum and will apply the limiter at full strength. Subsequentally, numerical dissipation
will smear the front across a few cells and the front width will occupy a wider stencil.
The discriminator, which works by comparing the curvature at a ﬁxed number of cells,
will fail to pick the widening front as a discontinuity, and will permit out of range values
to be mistaken for permissible smooth extrema.
In order to test the eﬀectiveness of the limiter, I have tried the 1D advection of a
square wave using the limited and unlimited WENO5 (5th order) coupled with RK2.
Figure 11.8 compares the negative minimum obtained with the limited (black) and unlimited (red) schemes; the xaxis represent time (the cone has undergone 4 rotations by
the end of the simulation). The diﬀerent panels show the result using 16, 32, 64 and 128
cells. The trend in all cases is similar for the unlimited scheme: a rapid growth of the
negative extremum before it reaches a quasisteady state. The trend for the limited case
is diﬀerent. Initially, the negative values are suppressed the black curves starting away
from time 0. This is initial period increases with better resolution. After the ﬁrst negative values appear, there is a rapid deterioration in the minimum value before reaching a
steady state. This steady state value decreases with the number of points, and becomes
negligeable for the 128 cell case. Finally, note that unlimited case produces a slightly
better minimum for the case of the 16 cells, but does not improve substantially as the
number of points is increased. For this experiment, the interval is the unit interval and
the hat proﬁle is conﬁned to x < 1/4; the time step is held ﬁx at ∆t = 1/80, so the
Courant number increases with the number of cells used. 11.7. UTOPIA 183 Figure 11.7: Advection of several Shchepetkin proﬁles. The black solid line refers to the
analytical solution, the red crosses to the WENO3 (RK2 timestepping), and the blue
stars to WENO5 with RK3. The WENO5 is indistiguishable from the analytical solution
for the narrow proﬁle I have repeated the experiments for the narrow proﬁle case (Shchepetkin’s proﬁle),
and conﬁrmed that the limiter is indeed able to supress the generation of negative value,
even for a resolution as low as 64 cells (the reference case uses 256 cells). The discriminator, however, allows a very slight and occasional increase of the peak value. By in
large, the limiter does a good job. The 2D cone experiments with the limiters are shown
in the Cone section. 11.7 Utopia The uniform third order polynomial interpolation algorithm was derived explicitly to be
a multidimension, twotime level, conservative advection scheme. The formulation is
based on a ﬁnite volume formulation of the advection equation:
(∆V T )t +
z ∂ ∆V F · n dS = 0 (11.62) where T is the average of T over the cell ∆V and F · n are the ﬂuxes passing through
the surfaces ∂ ∆V of the control volume. A further integral in time reduces the solution 184 CHAPTER 11. SPECIAL ADVECTION SCHEMES 0 0 10 10 −10 −10 10 10 −20 −20 10 10 0 100 200 300 0 200 300 0 0 100 100 200 300 0 10 10 −10 −10 10 10 −20 −20 10 10 0 100 200 300 Figure 11.8: Negative minima of unlimited (red) and limited (black) WENO scheme on
a square hat proﬁle. Top left 16 points, top right 32 points, bottom left 64 points, and
bottom rights 64 points. to the following:
T n+1 n =T + 1
∆V ∆t
0 ∂V F · n dS dt = 0 (11.63) A further deﬁnition will help us interpret the above formula. If we let the timeaverage
∆
ﬂux passing the surfaces bounding ∆V as F ∆t = 0 t F dt we end up with the following
twotime level expression:
T n+1 n =T + ∆t
∆V ∂V F · n dS (11.64) UTOPIA takes a Lagrangian point of view in tracing the ﬂuid particle crossing each
face. The situation is depicted in ﬁgure 11.9 where the particles crossing the left face
of a rectangular cell, is the area within the quadrilateral ABCD; this is eﬀectively the
contribution of edge AD to the boundary integral ∆t ∂ VAD F · n dS . UTOPIA makes
the assumption that the advecting velocity is locally constant across the face in space 11.7. UTOPIA 185 (j − 1, k + 1) (j, k + 1)
Et (j + 1, k + 1) A B (j − 1, k) (j, k) (j + 1, k) Ft D (j − 1, k − 1) (j − 1, k − 2) C
(j, k − 1) (j, k − 2) (j + 1, k − 1) (j + 1, k − 2) Figure 11.9: Sketch of the particles entering cell (j, k) through its left edge (j − 1 , k)
2
assuming positive velocity components u and v . 186 CHAPTER 11. SPECIAL ADVECTION SCHEMES and time; this amount to approximating the curved edges of the area by straight lines
as shown in the ﬁgure. The distance from the left edge to the straight line BC is u∆t,
and can be expressed as p∆x where p is the courant number for the xcomponent of the
velocity. Likewise, the vertical distance between point B and edge k + 1 is v ∆t = q ∆y ,
2
where q is the Courant number in the y direction.
We now turn to the issue of computing the integral of the boundary ﬂuxes; we will
illustrate this for edge AD of cell (j, k). Owing to UTOPIA’s Lagrangian estimate of the
ﬂux we have:
1
∆t
F · n dS =
T dx dy.
(11.65)
∆V ∂ VAD
∆x∆y ABCD
The right hand side integral is in area integral of T over portions of upstream neighboring
cells. Its evaluation requires us to assume a form for the spatial variations of T . Several
choices are available and Leonard et al Leonard et al. (1995) discusses several options.
UTOPIA is built on assuming a quadratic variations; for cell (j, k), the interpolation is:
1
T j +1,k + T j,k−1 + T j −1,k + T j,k+1 − 4T j,k
24
T j +1,k − T j −1,k
T j +1,k − 2T j,k + T j −1,k 2
ξ+
ξ
2
2
T j,k+1 − T j,k−1
T j,k+1 − 2T j,k + T j,k−1 2
η+
η.
2
2 Tj,k (ξ, η ) = T j,k −
+
+ (11.66) Here, ξ and η are scaled local coordinates:
ξ= x
y
−i η =
− j.
∆x
∆y (11.67) 1
so that the center of the box is located at (0, 0) and the left and right edges at (± 2 , 0),
respectively. The interpolation formula is designed such as to produce the proper cellaverages when the function is integrated over cells (j, k), (j ± 1, k) and (j, k ± 1). The area
integral in equation 11.65 must be broken into several integral: First, the area ABCD
stradles two cells, (j − 1, k) and (j − 1, k − 1), with two diﬀerent interpolation for T ; and
second, the trapezoidal area integral can be simpliﬁed. We now have 1
∆x∆y ABCD T dx dy = I1 (j, k) − I2 (j, k) + I2 (j, k − 1) (11.68) where the individual contributions from each area are given by:
I1 (j, k) =
I2 (j, k) = AEF D AEB Tj,k (ξ, η ) dη dξ = Tj,k (ξ, η ) dη dξ = 1
2 1
2 1
−u
2
1
2 1
−u
2
1
2 1
−u
2 ηAB (ξ ) Tj,k (ξ, η ) dη dξ (11.69) Tj,k (ξ, η ) dη dξ. (11.70) The equation for the line AB is:
ηAB (ξ ) = 1
1q
+
ξ−
2p
2 (11.71) 11.8. LAXWENDROFF FOR ADVECTION EQUATION 187 (fj,k+1 − 2fj,k + fj,k−1 ) − (fj,k − 2fj,k−1 + fj,k−2) 3
uv
24
fj,k−1 − 2fj,k + fj,k+1 2
uv
+
6
fj +1,k − 2fj,k + fj −1,k − fj +1,k−1 + 2fj,k−1 − fj −1,k−1 2
+
u
8
(fj +1,k − fj +1,k−1 ) − (fj,k − fj,k−1)
u
−
3
(fj,k−2 − 2fj,k−1 + fj,k ) + 2(fj +1,k − fj +1,k−1 )
+
12
2(fj,k+1 − fj,k−1) + (fj,k − fj −1,k ) − (fj,k−1 − fj −1,k−1)
+
uv
12
fj +1,k + fj,k fj +1,k − 2fj,k + fj −1,k
−
+
2
6
fj +1,k − 2fj,k + fj −1,k 2
fj +1,k − fj,k
u+
uu
(11.72)
−
2
6 Fj + 1 ,k =
2 Figure 11.10: Flux for the ﬁnite volume form of the utopia algorithm.
using the local coordinates of cell (j, k). Performing the integration is rather tedious; the
output of a symbolic computer program is shown in ﬁgure 11.10
A diﬀerent derivation of the UTOPIA scheme can be obtained if we consider the cell
values are function values and not cell averages. The ﬁnite diﬀerence form is then given
by the equation shown in ﬁgure 11.11. 11.8 LaxWendroﬀ for advection equation We explore the application of the LaxWendroﬀ procedure to compute highorder, twotime level approximation to the advection diﬀusion equation written in the form:
Tt + ∇ · (uT ) = 0. (11.74) The starting point is the time Taylor series expansion which we carry out to fourth order:
T n+1 = T n + ∆ t2
∆ t3
∆ t4
∆t
Tt +
Ttt +
Tttt +
Ttttt + O(∆t5 ).
1!
2!
3!
4! (11.75) The next step is the replacement of the timederivative above with spatial derivatives
using the original PDE. It is easy to derive the following identities:
Tt = −∇ · [uT ] (11.76) Ttt = −∇ · [ut T − u∇ · (uT )] (11.77) Ttt = −∇ · [ut T + uTt ] Tttt = −∇ · [utt T − 2ut ∇ · (uT ) − u∇ · (ut T ) + u∇ · (u∇ · (ut T ))] (11.78) 188 CHAPTER 11. SPECIAL ADVECTION SCHEMES fm,n−1 − 3fm,n−2 + 3fm,n+1 − fm,n 3
uv
24
fm,n−1 + fm,n+1 − 2fm,n 2
+
uv
6
−5fm,n−1 − 3fm+1,n−1 + 3fm,n+1 + 3fm+1,n + fm−1,n−1 − fm−1,n + fm,n−2 + fm,n
+
16
−fm+1,n + fm,n + fm+1,n−1 − fm,n−1
+
u
3
+fm+1,n + fm−1,n − fm+1,n−1 − 2fm,n + 2fm,n−1 − fm−1,n−1 2
+
u uv
8
+fm,n+1 − 3fm−1,n + 16fm,n + fm,n−1 + 9fm+1,n
+
24
fm+1,n − 2fm,n + fm−1,n 2
fm,n − fm+1,n
u+
uu
(11.73)
+
2
6 Fj + 1 ,k =
2 Figure 11.11: Flux of UTOPIA when variables represent function values. This is the
ﬁnite diﬀerence form of the scheme 11.9 2D Numerical experiments We present here a number of numerical experiments to illustrate the eﬀect of diﬀerent
discretization schemes on multidimensional advection of a passive tracer. The ﬂow is
a simple one and consist of a ﬂow rotating inside a square cavity with angular speed
π . The initial condition is the famous grooved cylinder experiment that include tracer
features that cannot align with a square grid, and sharp discontinuities. The preservation
of the sharp features and the extrema is the goal of this exercise. The sharp features
in the solution do not favor highorder scheme because they will lead to severe Gibbs
oscillations, whereas the diﬀusive scheme will smear the sharp gradients unrealistically.
There are six schemes considered here: donorcell (DC), fourthorder centered ﬁnite
diﬀerence (CD4), a FCTlimited CD4, a thirdorder UTOPIA scheme, a Universal limiter
based UTOPIAULim scheme, and a WENO scheme of order 5. There are 3 linear scheme
here (DC, CD4 and UTOPIA), and 3 nonlinear ones (FCTCD4, UTOPIAULim, and
WENO5). Only the CD4 is free of numerical diﬀusion whereas, DC and UTOPIA inject
second (laplacianlike diﬀusion) and fourthorder (biharmoniclike hyperdiﬀusion) spurious dissipation. The WENO5 scheme is also upwind biased but the oscillation control
is primarily due to the convex weighing of lower order stencils.
The numerical results are shown in ﬁgures 11.1211.17 in a convergence study form
for resolutions of 40, 80, 160, 320, and 640 cells in each direction. The lower right
plot shows the exact solution on the highest resolution grid (it does not reﬂect the
plotting distortion caused by the grid). A comparison among the diﬀerent scheme at the
intermediate resolution of 160 cells is shown in ﬁgure 11.18. The extrema of the ﬁeld
at the ﬁnal timestep are stamped for each experiment. The contour level shows are
equispaced between 0.1–1.0. We did not include the zerocontour as it leads to a lot of
noise because of tiny oscillations. Instead negative contours are shown in dashed black 11.9. 2D NUMERICAL EXPERIMENTS 189 lines for the two level 4hmin /5 and hmin /5.
The DC, UTOPIA, and UTOPIAULim schemes are singlestage twotime level schemes,
and hence sport the shortest computational cost. The CD4 and WENO5 ﬂavors were
integrated with an RK3TVD scheme. The latter was slightly modiﬁed to run the FCTversion of the CD4 scheme. In all cases the Courant number based on the corner velocity
was kept constant:
√
√
√ Nc
ωL 2/2T /Nt
2π
Umax ∆t
=
= 2π
=
= 0.06
(11.79)
C=
∆x
L/Nc
Nt
80
where Nc and Nt are the number of cells, and number of timesteps needed to perform
a single full rotation.
The Donor cell scheme shows a severe problem with excessive numerical diﬀusion.
The cylinder groove has been completely erased in the coarsest resolution run where one
sees an 77% loss of peak. The numerical damping improves with resolution, but even at
the highest one, the groove is not restored and there is still a 33% loss of peak. On the
other hand the solution of oscillationfree.
CD4 on the other hand is plagued by oscillations as no numerical dissipation is
available to dissipate the small scale noise. The oscillations persist from the coarsest
to the ﬁnest grid, and remain at the gridscale. Notice that in the unresolved case the
fourthorder scheme does a terrible job at capturing the solution features. Notice also
that the amplitude of the Gibbs oscillations of both under and overshoots remains quite
high irrespective of resolution.
The FCT corrected CD4 solution, 11.14, achieves its main goal of controlling the
Gibbs oscillations. Their amplitude now is of the order of machine precision. The shape
of the cylinder and its groove are still severely distorted in the underresolved cases,
but improve quickly with resolution. The severe damping of the DC scheme has been
eliminated, only 29% loss of peak at the coarsest resolution, and the resolving power of
the highorder scheme is apparent even at a resolution of 80 points.
The UTOPIA scheme is a thirdorder scheme with a hyperviscous numerical dissipation which acts much more vigorously on short scale waves then it does on longer scale
ones. The scheme however, is not oscillationfree but their amplitude and extent is much
less then that of CD4. The beneﬁts of scaleselective numerical diﬀusion are quite apparent for this example: control of Gibbs oscillation while preserving solution gradients
for longer times. At the intermediate resolution of 160 cells, the spurious extrema have
a much smaller amplitude than those of CD4, but larger than those of FCTCD4.
UTOPIAULim shows the good impact of combining a highorder scheme with a
limiter. At the resolved scale the solution is quite good and oscillationfree. At coarse
resolution the distortion is not compounded by gridscale noise. The universal limiter
does not belong to the class of FCT algorithm and has been designed to take multidimensionality into account.
The WENO5 scheme shown in 11.17 is not oscillationfree, but shows the impact of
resolution on the amplitude of these Gibbs oscillation: unlike CD4 where their amplitude
was independent of grid spacing, WENO5 leads to a decrease of Gibbs amplitude as ∆x
decreases. Notice also that the grooved cylinder is decently represented even at the
coarse resolution of 80 cells. The WENO5 is slightly upstreambiased and hence inject 190 CHAPTER 11. SPECIAL ADVECTION SCHEMES hmin =0.0006253 hmin =4.363e06 hmax =0.23878 hmax =0.39047 hmin =2.9875e10 hmin =1.5769e18 hmax =0.54917 hmax =0.63257 1 0.9 hmin =6.347e35 hmin =0 hmax =0.77175 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.12: Donor Cell with 40, 80, 160, 320, and 640 points 11.9. 2D NUMERICAL EXPERIMENTS 191 hmin =0.56894 hmin =0.39675 hmax =1.1613 hmax =1.644 hmin =0.52123 hmin =0.37856 hmax =1.3178 hmax =1.3952 1 0.9 hmin =0.3355 hmin =0 hmax =1.4186 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.13: Centered 4th order with 40, 80, 160, 320, and 640 points 192 CHAPTER 11. SPECIAL ADVECTION SCHEMES hmin =4.567e12 hmin =2.1219e19 hmax =0.71853 hmax =0.9012 hmin =7.3573e18 hmin =1.1295e17 hmax =0.99259 hmax =0.99999 1 0.9 hmin =1.3224e17 hmin =0 hmax =1 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.14: Centered 4th order with 40, 80, 160, 320, and 640 points 11.9. 2D NUMERICAL EXPERIMENTS 193 hmin =0.046307 hmin =0.053232 hmax =0.78887 hmax =1.0482 hmin =0.059122 hmin =0.15352 hmax =1.1221 hmax =1.1081 1 0.9 hmin =0.10015 hmin =0 hmax =1.1084 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.15: UTOPIA 40, 80, 160, 320, and 640 points 194 CHAPTER 11. SPECIAL ADVECTION SCHEMES hmin =3.2358e24 hmin =3.1356e36 hmax =0.74802 hmax =0.9478 hmin =1.9394e59 hmin =8.6597e154 hmax =1 hmax =1 1 0.9 hmin =8.3198e196 hmin =0 hmax =1 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.16: Limited UTOPIA 40, 80, 160, 320, and 640 points 11.9. 2D NUMERICAL EXPERIMENTS 195 hmin =6.254e06 hmin =0.020649 hmax =1.0583 hmax =1.0761 hmin =0.048594 hmin =0.00091886 hmax =1.0169 hmax =1.0479 1 0.9 hmin =8.2062e06 hmin =0 hmax =1.0005 hmax =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure 11.17: WENO5 with 40, 80, 160, 320, and 640 points 196 CHAPTER 11. SPECIAL ADVECTION SCHEMES a highlyscale selective numerical dissipation. This account for the limited extent of the
gridscale noise. Notice also that the sharp gradient is maitained as soon as it becomes
resolvable on the computational grid.
These ﬁnding are summarized in 11.18 where the diﬀerent scheme are compared for
the single resolution of 160 cells in each direction. It is clear that a highorder, upstreambiased scheme is almost optimal for this example as it strikes a good balance between
highorder and control of Gibbs oscillations. Limiters can enhance their performance by
removing unwanted noise. It should be pointed out that the cost of the scheme escalates
quite rapidly with the complexity of the algorithm. Although ﬁdelity to the analytical
solution is the over riding concern of the purists, practical considerations can trump the
use of expensive schemes. 11.9. 2D NUMERICAL EXPERIMENTS 197 hmin =2.9875e10 hmin =0.52123 hmax =0.54917 hmax =1.3178 hmin =0.059122 hmin =0.048594 hmax =1.1221 hmax =1.0169 hmin =7.3573e18 hmin =1.9394e59 hmax =0.99259 hmax =1 Figure 11.18: DC, CD4, UTOPIA, WENO5, FCTCENT4, LIMUTOPIA at 160 × 160 198 CHAPTER 11. SPECIAL ADVECTION SCHEMES Chapter 12 Fourier series
In this chapter we explore the issues arising in expressing functions as Fourier series. It
is extremely useful to recall some of the deﬁnitions of norms, inner products, and vector
spaces reviewed summarily in 15. 12.1 Continuous Fourier series As shown earlier, the set of functions
φk (x) = eikx , k = 0, ±1, ±2, . . . (12.1) forms an orthogonal basis function over the interval [−π, π ]:
π
−π φj φk dx = 2πδjk (12.2) where the overbar denotes the complex conjugate. The Fourier series of the function u
is deﬁned as:
Su = ∞ uk φk .
ˆ (12.3) k =−∞ It represents the formal expansion of u in the Fourier orthogonal system. The Fourier
coeﬃcients uk are:
ˆ
1π
u(x)e−ikx dx.
(12.4)
uk =
ˆ
2π −π
It is also possible to rewrite the Fourier series in terms of trigonometric functions by
using the identities:
cos θ = eiθ + e−iθ
eiθ − e−iθ
, sin θ =
,
2
2i (12.5) The Fourier series become:
Su = a0 + ∞ (ak cos kx + bk sin kx) k =1 199 (12.6) 200 CHAPTER 12. FOURIER SERIES The Fourier coeﬃcients of the trigonometric series are related to those of the complex
exponential series by
uk = ak − ibk .
ˆ
(12.7)
ˆ
If u is a realvalued functions, ak and bk are real numbers, and u−k = uk . Often it is
ˆ
unnessary to use the full Fourier expansion. If u is an even function, i.e. u(−x) = u(x)
then all the sine coeﬃcients, bk , are zero, and the series becomes what is called a cosineseries. Likewise, if the function u is odd, u(−x) = −u(x), bk = 0 and the expansion
becomes a sineseries. 12.2 Discrete Fourier series In practical applications, numerical methods based on Fourier series cannot be implemented in precisely the same way as suggested in the earlier section. For example, the
Fourier coeﬃcients of an arbitrary function may be too diﬃcult to calculate using equation 12.4, either the integral is too complicated to evaluate analytically, or the function
u is only known at a discrete set of points; thus an eﬃcient way must be found to convert the function u from physical space to spectral space. Furthermore, nonlinearities
can complicate signiﬁcantly the application of spectral methods. The key to overcoming
these diﬃculties is the use of a discrete Fourier series and its associated discrete Fourier
transform.
Consider the set of N points xj = 2πj/N , for j = 0, 1, . . . , N − 1. We deﬁne the
discrete Fourier coeﬀcients of a complex valued function u in 0 ≤ x ≤ 2π as
un =
˜ 1
N Notice that u±N/2 =
˜ N −1
j =0 uj e−inxj , n = −N/2, −N/2 + 1, . . . , N/2 − 1. N −1
∓i N
2
j =0 uj e 2πj
N = N −1
j
j =0 (−1) uj , (12.8) and so u N = u− N . The inver˜
˜
2 2 sion of the deﬁnition can be done easily, multiply equation 12.8 by einxk and sum the
resulting series over n to obtain:
N −1 un einxk =
˜ n=0 N −1
n=0 1
N N −1
j =0 uj ein(xk −xj ) = 1
N N −1
j =0 uj N −1 ein(xk −xj ) (12.9) n=0 The last sum can be written as as geometric series with factor ei(xk −xj ) . Its sum can
then be expressed analytically as:
N −1 ein(xk −xj ) = 1 + ei(xk −xj ) + ei2(xk −xj ) + . . . + ei(N −1)(xk −xj ) (12.10) n=0 = 1 − eiN (xk −xj )
1 − ei2π(k−j )
=
1 − ei2π(k−j )/N
1 − ei(xk −xj ) (12.11) There are two cases to consider: if k = j then all the terms in the series are equal to
1 and hence sum up to N , if k = j then the numerator in the last fraction is equal to
N−
0. Thus we can write n=01 ein(xk −xj ) = N δjk . The set of functions einx is said to be 12.2. DISCRETE FOURIER SERIES 201 discretely orthogonal. This property can be substituted in equation 12.9 to obtain the
inversion formula:
N −1 N −1 un einxk =
˜ n=0 12.2.1 uj δjk = uk (12.12) j =0 Fourier Series For Periodic Problems The structure of the discrete Fourier series can be justiﬁed easily for periodic problems.
In the continuous case they would take the form
∞ u(x) = un e−ikn x
ˆ (12.13) n=−∞ Here, and unlike the integral form, the wavenumbers k are not continuous but are quantizied on account of periodicity. If the domain is 0 ≤ x ≤ a then the wavenumbers are
given by
nπ
kn =
, n = 0, 1, 2, 3, . . .
(12.14)
a
In the discrete case the domain would be divide into an equallyspaced set of points
xj = j ∆x with ∆x = a/N , where N + 1 is the number of points. The discrete Fourier
series would then take the form
Nmax un e−ikn xj
ˆ u(xj ) = uj = (12.15) n=−Nmax where Nmax is the maximum wave mode that can be represented on the discrete grid.
Note that kn xj = 2πn jaN = 2πnj . Since the smallest wavelength is 2∆x the maximum
a
N
wavenumber is then
kmax = 2π
2πNmax
a
N
=
so that Nmax =
=
2∆x
a
2∆x
2 (12.16) Furthermore we have that
2π N
2 e−ik±Nmax xj = e∓i N j = e±iπj = (−1)j (12.17) Hence the two waves are identical and it is enough to retain the amplitude of one only.
The discrete Fourier series can then take the form:
N
−1
2 u(xj ) = uj = un e−i
ˆ 2πn
N (12.18) n=− N
2 We now have the parity between the N degrees of freedom in physical space uj and
the N degrees of freedom in Fourier space. Further manipulation can reduce the above
expression in the standard form presented earlier:
N
2 uj = −1 n=− N
2 un e−i
ˆ 2πnj
N (12.19) 202 CHAPTER 12. FOURIER SERIES
−1 = −i 2πnj
N un e
ˆ + = un e−i
ˆ −i 2πnj −i2πj
N e un e
ˆ + = 2π (n+N )j
−i
N un e
ˆ + = −i 2πnj
N un−N e
ˆ n= N
2 = N −1 N
−1
2 un e−i
ˆ 2πnj
N (12.21) N
−1
2 un e−i
ˆ 2πnj
N (12.22) n=0 n=− N
2
N −1 (12.20) n=0 n=− N
2
−1 2πnj
N n=0 n=− N
2
−1 N
−1
2 un e−i
˜ + N
−1
2 un e−i
ˆ 2πnj
N (12.23) n=0 2πnj
N (12.24) n=0 where the new tilded coeﬃcients are related to the old (hatted) coeﬃcients by
un = un
˜
ˆ
un = un−N
˜
ˆ N
2 0≤n≤ N −1
2
−1≤n≤N −1 (12.25) Sine transforms
For problems that have homogeneous Dirichlet boundary conditions imposed, one expands the solution in terms of sine functions:
uj = N −1 un sin
ˆ n=1 N −1
nπj ∆x N −1
nπj
nπxj
=
un sin
ˆ
=
un sin
ˆ
L
L
N
n=1
n=1 (12.26) It is easy to verify that u0 = uN = 0 no matter the values of un . To derive the inversion
ˆ
mπj
formula, multiply the above sum by sin N and sum over j to get:
N −1
j =1 mπj
uj sin
N = N −1
j =1 = N −1
n=1 = N −1
n=1 = N −1
n=1 un
ˆ N −1 n=1 N −1 un ˆ sin sin j =1 N −1 nπj
N sin mπj
N nπj
mπj sin
N
N (n + m)πj (n − m)πj
un ˆ
− cos
cos
2
N
N
j =1 (n−m)πj
(n+m)πj
(n+m)πj
un N −1 i (n−m)πj
ˆ + e−i N
− ei N
− e−i N (12.27)
eN
4
j =1 The terms in the inner series have the form Sk = r + r 2 + . . . + r k = r (1+ r + . . . + r k−1 ) = r r k+1 − r
r k+1 − 1
rk − 1
=
=
− 1 (12.28)
r−1
r−1
r−1 12.2. DISCRETE FOURIER SERIES 203 which for our trigonometric series become
S (p) = N −1 i pπj
N = e ei j =1 pπN
N pπ pπ pπ − ei N ei N − 1 = (−1)p − ei N
pπ ei N − 1 (12.29) with p = ±(n ± m). Note also that S (0) = (N − 1). Furthermore we have that
pπ S (p) + S (−p) = (−1)p − ei N pπ (−1)p − e−i N +
pπ
pπ
ei N − 1
e−i N − 1
(−1)p 2 cos pπ − 2 − 2 − 2 cos pπ
N
N
=
2 − 2 cos pπ
N
= −1 − (−1)p (12.30)
(12.31)
(12.32) We can now estimate the four diﬀerent sums:
S (n − m) + S (−n + m) − S (n + m) − S (−n − m) = (−1)n+m − (−1)n−m = 0 (12.33)
since if n ± m are even if n and m are both either odd or even, and n ± m is odd if one
is odd and the other even. For the case where m = n we have
S (n − m) + S (−n + m) − S (n + m) − S (−n − m) = 2(N − 1) + 1 + (−1)2n = 2N (12.34)
In compact notation we would write
S (n − m) + S (−n + m) − S (n + m) − S (−n − m) = 2N δnm (12.35) replacing the above expression in the discrete inner product we get
un =
ˆ 2
N N −1
j =1 uj sin πnj
N (12.36) 204 CHAPTER 12. FOURIER SERIES Chapter 13 Spectral Methods
13.1 Spectral Series The approximation of a function u by a series of the form
N u(x) = ui φj (x)
ˆ (13.1) j =1 raises several questions:
1. How many terms in the series should be kept to achieve a given error? Two related
questions are: how fast do the coeﬃcients ui decrease?, and what determines how
ˆ
fast these coeﬃcients decay?
2. How do we choose the basis functions φj (x)?
3. How do we compute the coeﬃcients uj ?
ˆ
There are 4 main concepts that are central to the following discussion.
1. The error in the approximation of a PDE comes from several sources, we however
assume that all of these decay to zero at rougly the same rate as N increases.
2. The second concept is anchored in Darboux’s principle, which states that the rate
of convergence of a series is determined by its distance to the closest singularity of
the function. Two functions that are diﬀerent by that share similar singularity (in
terms of location and strength) have spectral coeﬃcients that decay at the same
rate. By studying a few simple examples, we can learn a lot about the strength
and weaknesses of spectral methods.
3. From Darboux’s principle and a limited knowledge about the behavior of the function we can predict rates of convergence. Several rates are possible algebraic,
geometric, subgeometric and supergeometric.
4. From Darboux’s principle and the model functions we can produce rules of thumbs
to decide how to pick N .
205 206 CHAPTER 13. SPECTRAL METHODS 13.2 Fourier Series The Fourier series of a general function is
∞ u(x) = a0 + ∞ an cos nx + n=1 bn sin nx (13.2) n=1 where the cosine and sine Fourier coeﬃcients are:
a0 = 1
2π π an
bn u(x) dx,
−π = 1
π π u(x)
−π cos nx
sin nx dx, n = 1, 2, . . . (13.3) A more compact form to write the same equations makes use of complex notation, by
noting that einx = cos nx + i sin nx:
u(x) = ∞ cn einx , cn = n=−∞ 1
=
2π π
−π u(x)e−inx dx (13.4) The coeﬃcients of the two forms are related by:
c0 = a0 , cn = an + ibn n > 0
an − ibn n < 0 (13.5) Since the Fourier basis is periodic, it would be reasonable to assume that the basis is
solely useful for expanding periodic functions. This is partially true only. Fourier series
work best for periodic functions. However, the Fourier series will converge although very
slowly for quite arbitraty functions u(x).
Example 14 Consider the function u(x) = x, and let us ﬁnd its Fourier coeﬃcients.
Since the function is odd u(−x) = −x, all its cosine coeﬃcients are zero, while its sine
coeﬃcients are given by
bn = 1
π π
−π x sin nx dx = (−1)n+1 2
n (13.6) The amplitude of the coeﬃcients decrease linearly with the mode number, and hence the
series converges slowly as n increases.
Figure 13.1 shows a series of plot comparing the original function to the spectral
series with increasing truncation N . For low N the two curves are quite diﬀerent but the
diﬀerence decreases as N increases. Notice that the oscillations, away from the discontinuity, decrease with N while their amplitude remains constant near the discontinuity.
These oscillations are referred to as Gibbs oscillations produce an O(1) error near the
discontinuity since the function evaluates to π while the series sums up to 0. The damping of these oscillations requires a combination of techniques to damp their amplitudes,
such as “sum acceleration”, “ﬁltering”, and “reconstruction”. 13.2. FOURIER SERIES 207 N=1 N=5 N=9 N=18 N=36 N=72 Figure 13.1: Approximating the sawtooth function u(x) = x with a sinespectral series
with increasing cutoﬀ N . The solid black line is the original function and the blue line
the spectral approximation.
1 1 0.8 0.8 0.6 0.6 N=2 N=4 0.4 0.4 0.2 0.2 0 0 1 0 0.5 1 1.5 0.8 1 0
2 0.5 1 1.5 2 0.8 0.6 0.6 N=6 N=8 0.4 0.4 0.2 0.2 0 0
0 0.5 1 1.5 0
2 0.5 1 1.5 2 Figure 13.2: Approximating the function u(x) = max(0, sin(x) with a spectral series
with increasing cutoﬀ N . The solid black line is the original function and the blue line
the spectral approximation. The xaxis has been scaled by 1/π . 208 CHAPTER 13. SPECTRAL METHODS N=2 N=4 N=6 Figure 13.3: Approximating the function u(x) = 3/(5 − 4 cos x with a spectral series with
increasing cutoﬀ N . The solid black line is the original function and the blue line is the
spectral approximation. The xaxis has been scaled by 1/π .
Example 15 Our second example consists of expanding the function u(x) = max(0, sin(x)
on 0 ≤ x ≤ 2π . Notice that this function is continuous but not diﬀerentiable at x = 0
and x = π . The coeﬃcients can be calculated as:
n 1
a0 = π , an = −1(−(−1)
π n2 −1)
1
b1 = 2 , bn = 0 n>0
n>1 (13.7) The spectral series is rewritten as u = a0 + b1 sin nx + N=2 an cos nx and its graph
n
is shown in ﬁgure 13.2. One can visually notice that the series converges faster then the
previous example. The two curves almost overlap for N = 8, and the only noticeable
oscillations are near the kinks of the curve at x = 0 and 2π . The faster convergence is
in fact due to the quadratic, O(n−2 ), decrease of the spectral coeﬃcients, which in turn
is due to the increased smoothness of the function over the previous example.
Example 16 The function u(x) = 3/(5 − 4 cos x) is a periodic and inﬁnitely diﬀerentible
function on the interval x < π . It has the following Fourier expansion:
u(x) = 1 + 2 ∞ 2−n cos nx (13.8) n=1 Notice that for this case the Fourier coeﬃcients decrease geometrically fast with an+1 /an =
1/2. This rapid decrease in the amplitude of these coeﬃcients translate into a rapid
convergence of the series. The curves for the original function and for the spectral approximations are shown in ﬁgure 13.3 for several N , and the two curves overlap for
N = 6. The series exhibits exponential convergence as its coeﬃcients’ decay by 1/2 as n
increases.
For the previous cases the decay of the coeﬃcient is proportional to O(n−k ) so that
nk
k
an+1
∼
∼1− , n ≫k
k
an
(n + 1)
n (13.9) hence this ratio approaches 1 as n increases. In contrast the geometrically convergent
series has coeﬃcients ratio bounded away from 1. 13.2. FOURIER SERIES 209 0 10 10 −5 10 10 −10 10 0 10 10 1 0 10 20 30 40 50 60 70 10 0 −5 −10 n n Figure 13.4: loglog (left) and semilog plots of the decay of the Fourier coeﬃcients for
example 14 (black curve), example 15 (blue curve), and example 16 (red curve).
Figure 13.4 shows the decay of the Fourier coeﬃcients for the 3 examples presented
above on a loglog scale and on a semilog scale. We notice that functions with ﬁnite
regularity exhibit an algebraic convergence rate which translates into a straight line
on a loglog scale and a decreasing curve with upward concavity on a semilog scale.
Functions with inﬁnite regularity, like example 16 show faster decrease of their Fourier
coeﬃcients then the algebraic rates, and their graphs are straight lines on semilog scale
plots. The two additional curves shown illustrate the phenomena of subgeometric convergence, when expanding a function which is inﬁnitely diﬀerentiable but singular at
x = 0, and supergeometric convergence where the spectral coeﬃcients decrease faster
then exponentially as in e−n ln n . 13.2.1 Bounds on Fourier coeﬃcients To gain insight into the behavior of these coeﬃcients we return to their complex form:
cn = 1
2π π
−π u(x)e−inx dx (13.10) If the function u is diﬀerentiable on the interval x < π we can apply a single integration
by part to obtain:
cn = i
u(π ) − u(−π )
1
−
(−1)n i
2π
n
n π
−π u′ (x)e−inx dx (13.11) since einπ = e−inπ = cos(nπ = (−1)n . The process can be repeated again if u′ (x) is
continuous within the interval to get
cn = u′ (π ) − u′ (−π )
u(π ) − u(−π )
1
− (−1)n i2
−
(−1)n
2π
n
n2 i
n 2 π
−π u′′ (x)e−inx dx
(13.12) 210 CHAPTER 13. SPECTRAL METHODS The following equation can thus be obtained if the function u(x) is diﬀerentiable k times: (−1)n k
(−1)j u(j ) (π ) − u(j ) (−π )
cn =
2π
j =0 i
n j +1 1
−
2π i
n k +1 π
−π u(k+1) einx dx (13.13)
where u(j ) is the j th derivative of u. We note ﬁrst that if the u(j ) is continuous and
periodic we have u(j ) (π ) = u(j ) (−π ) for all j < k, and hence the individual entries in the
sum are all zero. The ﬁrst term not to vanish, i.e. for which the above condition does
not hold stops this integration process.
This series allows us to bound the coeﬃcients of the spectral expansion. If k is the
maximum number of time a function is diﬀentiable, and if u(j ) is periodic for j ≤ k − 2,
then the Fourier coeﬃcients decrease as
cn ∼ O 1
nk (13.14) If the function u is inﬁnitely diﬀerentiable and all its derivative are periodic, then the
process can be repeated an iniﬁnite number of times. This implies that the coeﬃcients
are decreasing faster then any ﬁnite power of n. This is the property of inﬁnite order or
exponential convergence. 13.3 Equal Error Assumptions Here we deﬁne the errors incurred in approximating the function u by a truncated series.
The error can be separated into several components which we deﬁne below:
1. Truncation Error ET (N ) is deﬁned to be the error made by neglecting all spectral
coeﬃcients an with n > N .
2. Discretization Error ED (N ) is the diﬀerence between the ﬁrst (N + 1) of the
exact solution and the corresponding terms as computed by a spectral or pseudospectral method using (N + 1) basis functions.
3. Interpolation Error EI (N ) is the error made by approximating a function by
(N + 1) term series w Chapter 14 Finite Element Methods
The discretization of complicated ﬂow domains with ﬁnite diﬀerence methods is quite
cumbersome. Their straightfoward application requires the ﬂow to occur in logically
rectangular domains, a constraint that severely limit their capabilities to simulate ﬂows
in realistic geometries. The ﬁnite element method was developed in large part to address
the issue of solving PDE’s in arbitrarily complex regions. Brieﬂy, the FEM method
works by dividing the ﬂow region into cells referred to as element. The solution within
each element is approximated using some form of interpolation, most often polynomial
interpolation; and the weights of the interpolation polynomials are adjusted so that
the residual obtained after applying the PDE is minimized. There are a number of
FE approaches in existence today; they diﬀer primarily in their choice of interpolation
procedure, type of elements used in the discretization; and the sense in which the residual
is minimized. Finlayson Finlayson (1972) showed how the diﬀerent approaches can be
uniﬁed via the perspective of the Mean Weighed Residual (MWR) method. 14.1 MWR Consider the following problem: ﬁnd the function u such that
L(u) = 0 (14.1) where L is a linear operator deﬁned on a domain Ω; if L is a diﬀerential operator,
appropriate initial and boundary conditions must be supplied. The continuum problem
as deﬁned in equation 14.1 is an inﬁnite dimensional problem as it requires us to ﬁnd
u at every point of Ω. The essence of numerical discretization is to turn this inﬁnite
dimensional system into a ﬁnite dimensional one:
N u=
˜ uj φ(x)
ˆ (14.2) j =0 Here u stands for the approximate solution of the problem, u are the N + 1 degrees
˜
ˆ
of freedom available to minimize the error, and the φ’s are the interpolation or trial
211 212 CHAPTER 14. FINITE ELEMENT METHODS functions. Equation 14.2 can be viewed as an expansion of the solution in term of a basis
function deﬁned by the functions φ. Applying this series to the operator L we obtain
L(˜) = R(x)
u (14.3) where R(x) is a residual which is diﬀerent then zero unless u is the exact solution of the
˜
equation 14.1. The degrees of freedom u can now be chosen to minimize R. In order to
ˆ
determine the problem uniquely, I can impose N + 1 constraints. For MWR we require
that the inner product of R with a N + 1 test functions vj to be orthogonal:
(R, vj ) = 0, j = 0, 1, 2, . . . , N. (14.4) Recalling the chapter on linear analysis; this is equivalent to saying that the projection of
R on the set of functions vj is zero. In the case of the inner product deﬁned in equation
15.13 this is equivalent to Ω Rvj dx = 0, j = 0, 1, 2, . . . , N. (14.5) A number of diﬀerent numerical methods can be derived by assigning diﬀerent choices
to the test functions. 14.1.1 Collocation If the test functions are deﬁned as the Dirac delta functions vj = δ(x−xj ), then constraint
14.4 becomes:
R(xj ) = 0
(14.6)
i.e. it require the residual to be identically zero on the collocation points xj . Finite
diﬀerences can thus be cast as a MWR with collocation points deﬁned on the ﬁnite
diﬀerence grid. The residual is free to oscillate between the collocation points where it is
pinned to zero; the oscillations amplitude will decrease with the number of of collocation
points if the residual is a smooth function. 14.1.2 Least Square ∂R
Setting the test functions to vj = ∂ uj is equivalent to minimizing the norm of the
ˆ
residual R 2 = (R, R). Since the only parameters available in the problem are uj , this
ˆ
2 . This minimum occurs for u such that
is equivalent to ﬁnding uj that minimize R
ˆ
ˆj ∂
∂ uj
ˆ R2 dx
2R Ω =0 (14.7) ∂R
dx
∂ uj
ˆ =0 (14.8) ∂R
∂ uj
ˆ =0 (14.9) Ω R, 14.2. FEM EXAMPLE IN 1D 14.1.3 213 Galerkin In the Galerkin method the test functions are taken to be the same as the trial functions,
so that vj = φj . This is the most popular choice in the FE community and will be the one
we concentrate on. There are varitions on the Galerkin method where the test functions
are perturbation of the trial functions. This method is usually referred as the PetrovGalerkin method. The perturbations are introduced to improve the numerical stability of
the scheme; for example to introduce upwinding in the solution of advection dominated
ﬂows. 14.2 FEM example in 1D We illustrate the application of the FEM method by focussing on a speciﬁc problem.
Find u(x) in x ∈ [0, 1] such that
∂2u
− λu + f = 0
∂x2 (14.10) subject to the boundary conditions
u(x = 0) = 0,
∂u
=q
∂x (14.11)
(14.12) Equation 14.11 is a Dirichlet boundary conditions and is usually referred to as an essential
boundary condition, while equation 14.12 is usually referred to as a natural boundary
conditions. The origin of these terms will become clearer shortly. 14.2.1 Weak Form In order to cast the equation into a residual formulation, we require that the inner
product with suitably chosen test functions v is zero:
1
0 ∂2u
− λu + f v dx = 0
∂x2 (14.13) The only condition we impose on the test function is that it is zero on those portions
of the boundary where Dirichlet boundary conditions are applied; in this case v (0) = 0.
Equation 14.13 is called the strong form of the PDE as it requires the second derivative
of the function to exist and be integrable. Imposing this constraint in geometrically
complex region is diﬃcult, and we seek to reformulate the problem in a “weak” form
such that only lower order derivatives are needed. We do this by integrating the second
derivative in equation 14.13 by part to obtain:
1
0 ∂u ∂v
+ λuv dx =
∂x ∂x 1 f v dx + v
0 ∂u
∂x 1 −v ∂u
∂x (14.14)
0 The essential boundary conditions on the left edge eliminates the third term on the right
hand side of the equation since v (0) = 0, and the Neumann boundary condition at the 214 CHAPTER 14. FINITE ELEMENT METHODS right edge can be substituted in the second term on the right hand side. The ﬁnal form
is thus:
1 ∂ u ∂v
1
+ λuv dx =
f v dx + qv (1)
(14.15)
∂x ∂x
0
0
For the weak form to be sensible, we must require that the integrals appearing in the
formulation be ﬁnite. The most severe restriction stems from the ﬁrst order derivatives
appearing on the left hand side of 14.15. For this term to be ﬁnite we must require that
∂v
the functions ∂u and ∂x be integrable, i.e. piecewise continuous with ﬁnite jump discon∂x
tinuities; this translates into the requirement that the functions u and v be continuous,
the socalled C0 continuity requirement. 14.2.2 Galerkin form The solution is approximated with a ﬁnite sum as:
N u(x) = ui φi
ˆ (14.16) i=0 and the test functions are taken to be v = φj , j = 1, 2, . . . , N . The trial functions φi
must be chosen such that φi>0 (0) = 0, in accordance with the restriction that v (0) = 0.
We also set, without loss of generality, φ0 (0) = 1, the ﬁrst term of the series is then
nothing but the value of the function at the edge where the Dirichlet condition is applied:
u(0) = u0 . The substitution of the expansion and test functions into the weak form yield
ˆ
the following N system of equations in the N + 1 variables ui :
ˆ
N 1 i=0 0 ∂ φi ∂φj
+ λφi φj dx ui =
ˆ
∂x ∂x 1
0 f φj dx + qφj (1) (14.17) In matrix form this can be rewritten as
N 1 Kji ui = bj , Kji =
i=0 0 ∂φj ∂φi
+ λφi φj dx, bj =
∂x ∂x 1
0 f φj dx + qφj (1) (14.18) Note that the matrix K is symmetric: Kji = Kij , so that only half the matrix entries
need to be evaluated. The Galerkin formulation of the weak variational statement 14.15
will always produce a symmetric matrix regardless of the choice of expansion function;
the necessary condition for the symmetry is that the left hand side operator in equation
14.15 be symmetric with respect to the u and v variables. The matrix K is usually
referred to as the stiﬀness matrix, a legacy term dating to the early application of the
ﬁnite element method to solve problems in solid mechanics. 14.2.3 Essential Boundary Conditions The series has the N unknowns u1≤i≤N , thus the matrix equation above must be modiﬁed
ˆ
to take into account the boundary condition. We do this by moving all known qunatities
to the right hand side, and we end up with the following system of equations:
N
i=1 Kji ui = cj , cj = bj − Kj 0 u0 , j = 1, 2, . . . , N (14.19) 14.2. FEM EXAMPLE IN 1D 215 Had the boundary condition on the right edge of the domain been of Dirichlet type,
we would have to add the following restrictions on the trial functions φ2≤i≤N −1 (1) = 0.
The imposition of Dirichlet conditions on both sides is considerably simpliﬁed if we
further request that φ0 (1) = φN (0) = 0 and φ0 (0) = φN (1) = 1. Under these conditions
u0 = u(0) = u0 and uN = u(1) = uN . We end up with the following (N − 1) × (N − 1)
ˆ
ˆ
system of algebraic equations
N −1
i=1 14.2.4 Kji ui = cj , cj = bj − Kj 0 u0 − KjN uN , j = 1, 2, . . . , N − 1 (14.20) Choice of interpolation and test functions To complete the discretization scheme we need to specify the type of interpolation functions to use. The choice is actually quite open save for the restriction on using continuous functions (to integrate the ﬁrst order derivative terms), and imposing the Dirichlet
boundary conditions. There are two aspects to choosing the test functions: their locality
and their interpolation properties.
If the functions φi are deﬁned over the entire domain, they are termed global expansion functions. Such functions are most often used in spectral and pseudospectral
methods. They provide a very accurate representation for functions that are smooth; in
fact the rate of convergence increases faster then any ﬁnite power of N if the solution is
inﬁnitely smooth, a property known as exponential convergence. This property is lost if
the solution has ﬁnite continuity. The main drawback of global expansion functions is
that the resulting matrices are full and tightly couple all degrees of freedom. Furthermore, the accurate representation of local features, such as fronts and boundary layers,
requires long series expasions with substantial increase in the computational cost.
Finite element methods are based on local expansion functions: the domain is divided into elements wherein the solution is expanded into a ﬁnite series. The functions φi
are thus nonzero only within one element, and zero elsewhere. This local representation
of the function is extremely useful if the solution has localized features such as boundary
layers, local steep gradient, etc... The resulting matrices are sparser and hence more eﬃcient solution schemes become possible. The most popular choice of expansion function
is the linear interpolation function, commonly referred to the hat functions which we
will explore later on. Higher order expansion are also possible, in particular the spectral
element method chooses expansion that are high order polynomial within each element.
The nature of the expansion function refers to the nature of the expansion coeﬃcients.
If a spectral representation is chosen, then the unknowns become the generalized Fourier
(or spectral) coeﬃcients. Again this is a common choice for spectral methods. The
most common choice of expansion functions in ﬁnite element methods are Lagrangian
interpolation functions, i.e. functions that interpolated the solution at speciﬁed points
xj also referred to as collocation points; in FEM these points are also referred to as
nodes. Lagrangian interpolants are chosen to have the following property:
φj (xi ) = δij (14.21) where δij is the Kronecker delta. The interpolation function φj is zero at all points xi=j ;
at point xj , φj (xj ) = 1. Each interpolation function is associated with one collocation 216 CHAPTER 14. FINITE ELEMENT METHODS φ0 φ1 φ2 u0
ˆ u1
ˆ u2
ˆ
$
$
$$
$$
$$
$$
$$
$$
$$$
$$$
s
$$
s $
$
s Figure 14.1: 2element discretization of the interval [0, 1] showing the interpolation functions
point. If our expansion functions are Lagrange interpolants, then the coeﬃcients ui
ˆ
represent the value of the function at the collocation points xj :
N u(xj ) = uj = ui φ(xj ) = uj , j = 0, 1, . . . , N
ˆ
ˆ (14.22) i=0 We will omit the circumﬂex accents on the coeﬃcients whenever the expansion functions
are Lagrangian interpolation functions. The use of Lagrangian interpolation simpliﬁes
the imposition of the C 0 continuity requirement, and the function values at the collocation points are obtained directly without further processing.
There are other expansion functions in use in the FE community. For example,
Hermitian interpolation functions are used when the solution and its derivatives must be
continuous across element boundaries (the solution is then C 1 continuous); or Hermitian
expansion is used to model inﬁnite elements. These expansion function are usually
reserved for special situations and we will not address them further.
In the following 3 sections we will illustrate how the FEM solution of equation 14.15
proceeds. We will approach the problem from 3 diﬀerent perspectives in order to highlight
the algorithmic steps of the ﬁnite element method. The ﬁrst approach will consider a
small size expansion for the approximate solution, the matrix equation can then be
written and inverted manually. The second approach repeats this procedure using a
longer expansion, the matrix entries are derived but the solution of the larger system
must be done numerically. The third approach considers the same large problem as
number two above; but introduces the local coordinate and numbering systems, and the
mapping between the local and global systems. This local approach to constructing the
FE stiﬀness matrix is key to its success and versatility since it localizes the computational
details to elements and subroutines. A great variety of local ﬁnite element approximations
can then be introduced at the local elemental level with little additional complexity at
the global level. 14.2.5 FEM solution using 2 linear elements We illustrate the application of the Galerkin procedure for a 2element discretization of
1
the interval [0, 1]. Element 1 spans the interval [0, 2 ] and element 2 the interval [ 1 , 1]
2
and we use the following interpolation procedure:
u(x) = u0 φ0 (x) + u1 φ1 (x) + u2 φ2 (x) (14.23) 14.2. FEM EXAMPLE IN 1D 217 where the interpolation functions and their derivatives are tabulated below
φ0 (x)
φ1 (x)
φ2 (x)
1
0 ≤ x ≤ 2 1 − 2x
2x
0
1
≤x≤1
0
2(1 − x) 2x − 1
2 ∂φ0
∂x ∂φ1
∂x ∂φ2
∂x −2
0 2
−2 0
2 (14.24) and shown in ﬁgure 14.1. It is easy to verify that the φi are Lagrangian interpolation
functions at the 3 collocations points x = 0, 1/2, 1, i.e. φi (xj ) = δij . Furthermore, the
expansion functions are continuous across element interfaces, so that the C 0 continuity
requirement is satisﬁed), but their derivates are discontinuous. It is easy to show that the
interpolation 14.23 amounts to a linear interpolation of the solution within each element.
Since the boundary condition at x = 0 is of Dirichlet type, we need only test with
functions that satisfy v (0) = 0; in our case the functions φ1 and φ2 are the only candidates. Notice also that we have only 2 unknowns u1 and u2 , u0 being known from
the Dirichlet boundary conditions; thus only 2 equations are needed to determine the
solution. The matrix entries can now be determined. We illustrate this for two of the
entries, and assuming λ is constant for simplicity:
1 K10 = 0 ∂ φ1 ∂φ0
+ λφ1 φ0
∂x ∂x 1
2 dx =
0 [−4 + λ(2x − 4x2 )] dx = −2 + λ
12 (14.25) Notice that the integral over the entire domain reduces to an integral over a single element
because the interpolation and test functions φ0 and φ1 are nonzero only over element 1.
This property that localizes the operations needed to build the matrix equation is key
to the success of the method.
The entry K11 requires integration over both elements:
1 K11 = ∂ φ1 ∂φ1
+ λφ1 φ1
∂x ∂x 0
1
2 = 1 [4 + λ4x2 ] dx + 1
2 0 = 2+ 2λ
12 + 2+ 2λ
12 dx (14.26) [4 + λ4(1 − x)2 ] dx
=4+ 4λ
12 (14.27)
(14.28) The remaining entries can be evaluated in a similar manner. The ﬁnal matrix equation
takes the form:
−2 +
0 λ
12 4 + 4λ
12
λ
−2 + 12 λ
−2 + 12
2λ
2 + 12 u0 u1 =
u2 b1
b2 (14.29) Note that since the value of u0 is known we can move it to the right hand side to obtain
the following system of equations:
4 + 4λ
12
λ
−2 + 12 λ
−2 + 12
2λ
2 + 12 u1
u2 = b1 + 2 −
b2 λ
12 u0 (14.30) 218 CHAPTER 14. FINITE ELEMENT METHODS whose solution is given by
u1
u2 = 2 + 2λ
12
λ
2 − 12 1
∆ λ
2 − 12
4 + 4λ
12 b1 + 2 −
b2 λ
12 u0 (14.31) λ
λ
where ∆ = 8(1 + 12 )2 − ( 12 − 2)2 is the determinant of the matrix. The only missing
piece is the evaluation of the right hand side. This is easy since the function f and the
ﬂux q are known. It is possible to evaluate the integrals directly if the global expression
for f is available. However, more often that not, f is either a complicated function, or is
known only at the collocation points. The interpolation methodology that was used for
the unknown function can be used to interpolate the forcing functions and evalute their
associated integrals. Thus we write:
1 bj = f φj dx + qφj (1) 0 (14.32) 12 = (fi φi )φj dx + qφj (1) (14.33) 0 i=0
2
1 =
0 i=0 1
12 = φi φj dx fi + qφj (1) 141
012 The ﬁnal solution can thus be written as:
u1
u2 = 1
∆ 2+
2− 2λ
12
λ
12 2−
4+ f0 f1 +
f2
f0 +4f1 +f2
+
12
f 1+2f2
12 λ
12
4λ
12 (14.34)
0
q 2− (14.35) λ
12 +q u0 (14.36) If u0 = 0, λ = 0 and f = 0, the analytical solution to the problem is u = qx. The
ﬁnite element solution yields:
u1
u2 = 1
4 22
24 0
q = q
2 q (14.37) which is the exact solution of the PDE. The FEM procedure produces the exact result
because the solution to the PDE is linear in x. Notice that the FE solution is exact at
the interpolation points x = 0, 1/2, 1 and inside the elements.
If f = −1, and the remaining parameters are unchanged the exact solution is
quadratic ue = x2 /2 + (q − 1)x, and the ﬁnite element solution is
u1
u2 = 1
4 22
24 −1
2 q− 1
4 = 4q −3
8
q−1
2 (14.38) Notice that the FE procedure yields the exact value of the solution at the 3 interpolation
points. The errors committed are due to the interpolation of the function within the 14.2. FEM EXAMPLE IN 1D 219 0.5 0.7
0.6 f=0 0.4 0.5
0.3 0.4 0.2 0.3 f=−x 0.2
0.1
0 0.1
0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 f=−x2 0.4 0.2 0 0 0.2 Figure 14.2: Comparison of analytical (solid line) and FE (dashed line) solutions for
the equation ux x + f = 0 with homogeneous Dirichlet condition on the left edge and
Neumann condition ux = 1 on the right edge. The circles indicate the location of the
interpolation points. Two linear ﬁnite elements are used in this example. 220 CHAPTER 14. FINITE ELEMENT METHODS elements; the solution is quadratic whereas the FE interpolation provide only for a linear
variation.
For f = −x, the exact solution is ue = x3 /6 + (q − 1/2)x, and the FE solution is:
u1
u2 = 1
4 −1
4 22
24 q− 5
24 = 24q −11
48
q−1
3 (14.39) The solution is again exact at the interpolation points by in error within the element due
to the linear interpolation. This fortuitous state of aﬀair is due to the exact evaluation
of the forcing term f which is also exactly interpolated by the linear functions.
For f = −x2 , the exact solution is u = x4 /12 + (q − 1/2)x, and the FE solution is:
u1
u2 = 1
4 −1
12 22
24 q− 9
48 = 48q −10
96
q − 22
96 (14.40) This time the solution is in error at the interpolation points also. Figure 14.2 compare
the analytical and FE solutions for the 3 cases after setting q = 1. 14.2.6 FEM solution using N linear elements φ0 rr
rr
u
ru x0 ∆ x0 x
1 φj −1
... φj φj +1
$
$
$$$$
$$$$
$$$
$$$
u$
$
u $
$
u xj − 1 ∆ xj xj ∆xj +1 xj +1 φN ... ¨¨
¨
¨
¨
u
u xN − 1 ∆ xN xN Figure 14.3: An element partition of the domain into N linear elements. The element
edges are indicated by the ﬁlled dots.
In order to decrease the error further for cases where f is a more complex function,
we need to increase the number of elements. This will increase the size of the matrix
system and the computational cost. We go through the process of developing the stiﬀness
equations for this system since it will be useful for the understanding of general FE
concepts. Suppose that we have divided the interval into N elements (not necessarily of
equal size), then interpolation formula becomes:
N u(x) = ui φi (x) (14.41) i=0 Element number j , shown in ﬁgure 14.3, spans the interval [xj −1 , xj ], for j = 1, 2, . . . , N ;
its left neighbor is element j − 1 and its right number is element j + 1. The length of
each element is ∆xj = xj − xj −1 . The linear interpolation function associated with node 14.2. FEM EXAMPLE IN 1D 221 j is
φj (x) = 0
x − x j −1
x j − x j −1
xj +1 −x
xj +1 −xj 0 x < xj −1
xj −1 ≤ x ≤ xj (14.42) xj ≤ x ≤ xj +1
xj +1 < x Let us now focus on building the stiﬀness matrix row by row. The j th row of K
corresponds to setting the test function to φj . Since the function φj is nonzero only
on the interval [xj −1 , xj +1 ], the integral in the stiﬀness matrix reduces to an integration
over that interval. We thus have:
xj +1 ∂ φ ∂φ
j
i
(14.43)
+ λφi φj dx,
Kji =
∂x ∂x
x j −1
bj = xj +1
x j −1 f φj dx + qφj (1) (14.44) Likewise, the function φi is nonzero only on elements i and i + 1, and hence Kji = 0
unless i = j − 1, j, j + 1; the system of equation is hence tridiagonal. We now derive
explicit expressions for the stiﬀness matrix entries for i = j, j ± 1.
xj Kj,j −1 = x j −1 ∂ φj ∂φj −1
+ λφj −1 φj
∂x ∂x dx, (14.45) xj x − xj − 1 xj − x
1
+λ
2
(xj − xj −1 )
xj − xj −1 xj − xj −1
x j −1
∆xj
1
+λ
=−
∆ xj
6
= − dx, (14.46)
(14.47) The entry for Kj,j +1 can be deduced automatically by using symmetry and applying the
above formula for j + 1; thus:
Kj,j +1 = Kj +1,j = − 1
∆xj +1
+λ
∆xj +1
6 (14.48) The sole entry remaining is i = j ; in this case the integrals spans the elements i and i + 1
xj Kj,j =
=
= x j −1 1
∆x2
j ∂ φj ∂φj
+ λφj φj
∂x ∂x
xj
x j −1 dx + xj +1
xj 1 + λ(x − xj −1 )2 dx, + 1
2∆xj
+λ
∆ xj
6 + ∂ φj ∂φj
+ λφj φj
∂x ∂x 1
∆x2+1
j xj +1
xj dx, (14.49) 1 + λ(xj +1 − x)2 (14.50)
dx, 2∆xj +1
1
+λ
∆xj +1
6 Note that all entries in the matrix equations are identical except for the rows associated
with the end points where the diagonal entries are diﬀerent. It is easy to show that we
must have:
1
2∆x1
K0,0 =
+λ
(14.51)
∆x1
6
2∆xN
1
+λ
(14.52)
KN,N =
∆xN
6 222 CHAPTER 14. FINITE ELEMENT METHODS
The evaluation of the right hand sides leads to the following equations for bj :
bj = xj +1
x j −1
j +1 = f φj dx + qφj (1)
xj +1 i=j −1 xj −1 φi φj dxfi + qφN (1)δN j (14.53)
(14.54) 1
[∆xj fj −1 + 2(∆xj + ∆xj +1 )fj + ∆xj +1 fj +1 ] + qφN (1)δN j (14.55)
6
Again, special consideration must be taken when dealing with the edge points to account
for the boundary conditions properly. In the present case b0 is not needed since a Dirichlet
boundary condition is applied on the left boundary. On the right boundary the right
hand side term is given by:
= 1
[∆xj fN −1 + 2(∆xN fj ] + q
(14.56)
6
Note that the ﬂux boundary condition aﬀects only the last entry of the right hand side.
If the grid spacing is constant, a typical of the matrix equation is:
bN = 1
λ∆ x
2λ∆x
λ∆ x
1
1
+
+
+
uj − 1 + 2
uj + −
uj +1 =
∆x
6
∆x
6
∆x
6
∆x
(fj −1 + 4fj + fj +1 ) (14.57)
6
For λ = 0 it is easy to show that the left hand side reduces to the centered ﬁnite diﬀerence
approximation of the second order derivative. The ﬁnite element discretization produces
a more complex approximation for the right hand side involving a weighing of the function
at several neighboring points.
− 14.2.7 Local stiﬀness matrix and global assembly We note that in the previous section we have built the stiﬀness matrix by constantly
referring to a global node numbering system and a global coordinate system across all
elements. Although this is practical and simple in onedimensional problems, it becomes
very tedious in two and three dimensions, where elements can have arbitrary orientation
with respect to the global coordinate system. It is thus useful to transform the computations needed to a local coordinate system and a local numbering system in order to
simplify/automate the building of the stiﬀness matrix. We illustrate these local entities
in the onedimensional case since they are easiest to grasp in this setting.
For each element we introduce a local coordinate system that maps the element j
deﬁned over xj −1 ≤ x ≤ xj into −1 ≤ ξ ≤ 1. The following linear map is the simplest
transformation that accomplishes that:
x − xj −1
−1
(14.58)
ξ=2
∆xj
This linear transformation maps the point xj −1 into ξ = −1 and xj into ξ = 1; its inverse
is simply
ξ+1
+ xj −1
(14.59)
x = ∆ xj
2 14.2. FEM EXAMPLE IN 1D 223
h1 (ξ ) uj
1 h2 (ξ )
$$
$$
$$
$
u uj
$
u $$
2 ∆ xj ξ1 ξ2 Figure 14.4: Local coordinate system and local numbering system
We also introduce a local numbering system so the unknown can be identiﬁed locally.
The superscript j , whenever it appears, indicate that a local numbering system is used to
refer to entities deﬁned over element j . In the present instance the uj refers to the global
1
unknown uj −1 and uj refers to the global unknown uj . Finally, the global expansion
2
functions, φj are transformed into local expansion functions so that the interpolation of
the solution u within element j is:
uj (ξ ) = uj h1 (ξ ) + uj h2 (ξ )
1
2 (14.60) where the functions h1,2 are the local Lagrangian functions
h1 (ξ ) = 1−ξ
1+ξ
, h2 (ξ ) =
,
2
2 (14.61) It is easy to show that h1 should be identiﬁed with the right limb of the global function
φj −1 while h2 should be identiﬁed with the left limb of global function φj (x).
The operations that must be carried out in the computational space include diﬀerentiation and integration. The diﬀerentiation in physical space is evaluated with the help
of the chain rule:
∂uj ∂ξ
∂uj 2
∂uj
=
=
(14.62)
∂x
∂ξ ∂x
∂ξ ∆xj
∂ξ
where ∂x is the metric of mapping element j from physical space to computational space.
For the linear mapping used here this metric is constant. The derivative of the function
in computational space is obtained from diﬀerentiating the interpolation formula 14.60: ∂uj
∂ξ = uj
1
= ∂h1
∂h2
+ uj
2
∂ξ
∂ξ (14.63) uj − uj
2
1
2 (14.64) For the linear interpolation functions used in this example, the derivative is constant
throught the element.
We know introduce the local stiﬀness matrices which are the contributions of the
local element integration to the global stiﬀness matrix:
j
Km,n = xj
x j −1 ∂ hm ∂hn
+ λhm (ξ )hn (ξ )
∂x ∂x dx, (m, n) = 1, 2 (14.65) Notice that the local stiﬀness matrix has a small dimension, 2 × 2 for the linear interpolation function, and is symmetric. We evaluate these integrals in computational space 224 CHAPTER 14. FINITE ELEMENT METHODS j
j
by breaking them up into 2 pieces Dm,n and Mm,n deﬁned as follows:
xj 1 ∂hm ∂hn
dx =
∂x ∂x ∂hm ∂hn
x j −1
−1 ∂ξ ∂ξ
1
xj
∂x
dξ
hm hn
hm hn dx =
∂ξ
−1
x j −1 j
Dm,n =
j
Mm,n = ∂ξ
∂x 2 ∂x
dξ
∂ξ (14.66)
(14.67) The integrals in physical space have been replaced with integrals in computational space
in which the metric of the mapping appears. For the linear mapping and interpolation
function, these integrals can be easily evaluated:
Mj = ∆xj
2 1
−1 (1 − ξ )2 (1 − ξ 2 )
(1 − ξ 2 ) (1 + ξ )2 dξ = ∆xj
6 21
12 (14.68) 1 −1
−1 1 (14.69) Similarly, the matrix D j can be shown to be:
Dj = 1
2∆xj 1
−1 1 −1
−1 1 dξ = 1
∆ xj The local stiﬀness matrix is K j = D j + λM j . The matrix M j appears frequently in
FEM, it is usually identiﬁed with a timederivative term (absent here), and is referred
to as the mass matrix.
Having derived expressions for the local stiﬀness matrix, what remains is to map
them into the global stiﬀness matrix. The following relationships hold between the
global stiﬀness matrix and the local stiﬀness matrices:
j
Kj,j −1 = K2,j j
j
Kj,j = K2,2 + K1,+1
1 Kj,j +1 = j
K1,+1
2 (14.70)
(14.71)
(14.72) The left hand sides in the above equations are the global entries while those on the
right hand sides are the local entries. The process of adding up the local contribution is
called the stiﬀness assembly. Note that some global entries require contributions from
diﬀerent elements.
In practical computer codes, the assembly is eﬀected most eﬃciently by keeping
track of the map between the local and global numbers in an array: imap(2,j) where
imap(1,j) gives the global node number of local node 1 in element j , and imap(2,j)
gives the global node number of local node 2 in element j . For the onedimensional case
a simple straightforward implementation is shown in the ﬁrst loop of ﬁgure 14.5 where P
stands for the number of collocation points within each element. For linear interpolation,
P = 2. The scatter, gather and assembly operations between local and global nodes can
now be easily coded as shown in the second, and third loops of ﬁgure 14.5. 14.2.8 Quadratic Interpolation With the local approach to stiﬀness assembly, it is now simple to deﬁne more complicated
local approximations. Here we explore the possibility of using quadratic interpolation to 14.2. FEM EXAMPLE IN 1D integer, parameter :: N=10
integer, parameter :: P=3
integer :: Nt=N*(P1)+1
integer :: imap(P,N)
real*8 :: ul(P,N),vl(P,N)
real*8 :: u(Nt), v(Nt)
real*8 :: Kl(P,P,N)
real*8 :: K(Nt,Nt) 225 !
!
!
!
!
!
!
! number of elements
number of nodes per element
total number of nodes
connectivity
local variables
global variables
local stiffness matrix
global stiffness matrix !
Assign Connectivity in 1D
do j = 1,N
! element loop
do m = 1,P
! loop over collocation points
imap(m,j) = (j1)*(P1)+m ! assign global node numbers
enddo
enddo
!
Gather/Scatter operation
do j = 1,N
! element loop
do m = 1,P
! loop over local node numbers
mg = imap(m,j)
! global node number of node m in element j
ul(m,j) = u(mg)
! local gathering operation
v(mg) = vl(m,j)
! local scattering
enddo
enddo
!
Assembly operation
K(1:Nt,1:Nt) = 0
! global stiffness matrix
do j = 1,N
! element loop
do n = 1,P
ng = imap(n,j)
! global node number of local node n
do m = 1,P
mg = imap(m,j)
! global node number of local node m
K(mg,ng) = K(mg,ng) + Kl(m,n,j)
enddo
enddo
enddo
Figure 14.5: Gather, scatter and stiﬀness assembly codes. 226 CHAPTER 14. FINITE ELEMENT METHODS improve our solution. The local interpolation takes the form
uj (ξ ) = uj h1 (ξ ) + uj h2 (ξ ) + uj h3 (ξ )
1
2
3
1−ξ
h1 (ξ ) = −ξ
2
h2 (ξ ) = 1 − ξ 2
1+ξ
h3 (ξ ) = ξ
2 (14.73)
(14.74)
(14.75)
(14.76) It is easy to verify that hi (ξ ) are Lagrangian interpolants at the collocation points ξi =
−1, 0, 1. These functions are shown in top right panel of ﬁgure 14.6. Notice that there
are now 3 degrees of freedom per elements, and that the interpolation function associated
with the interior node does not interact with the interpolation functions deﬁned in other
elements. The local matrices can be evaluated analytically: 7 −8
1
4 2 −1
1
∆ xj j
j
2 , D =
M= −8 16 −8 , 2 16
30
3∆xj
1 −8
7
−1 2
4 (14.77) The assembly procedure can now be done as before with the proviso that the local node
numbers m runs from 1 to 3. In the present instance the global system of equation is
pentadiagonal and is more expensive to invert then the tridiagonal system obtained with
the linear interpolation functions. One would expect improved accuracy, however. 14.2.9 Spectral Interpolation Generalizing the approach to higher order polynomial interpolation is possible. As the
degree of the interpolating polynomial increases, however, the wellknown Runge phenomenon rears its ugly head. This phenomenon manifests itself in oscillations near the
edge of the interpolation interval. This can be cured by a more judicial choice of the
collocation points. This is the approach followed by the spectral element method, where
the polynomial interpolation is still cast in terms of high order Lagrangian interpolation polynomials but the collocation points are chosen to be the GaussLobatto roots
of special polynomials. The most common polynomials used are the Legendre polynomials since their GaussLobatto roots possess excellent interpolation and quadrature
properties. The Legendre spectral interpolation takes the form
P uj (ξ ) = uj hm (ξ )
m (14.78) m=1 hm (ξ ) = P
−(1 − ξ 2 )L′ −1 (ξ )
ξ − ξn
P
=
,
P (P − 1)LP −1 (ξm )(ξ − ξm ) n=1,n=m ξm − ξn m = 1, 2, . . . , P.
(14.79) LP −1 denotes the Legendre polynomial of degree (P − 1) and L′ −1 denotes its derivative.
P
The P collocation points ξn are the P GaussLobatto roots of the Legendre polynomials
of degree P − 1, i.e. they are the roots of the equation:
2
(1 − ξn )L′ −1 (ξn ) = 0,
P (14.80) 14.2. FEM EXAMPLE IN 1D 227 1 h2
0.8 1 1 h2
2 0.8 0.6 0.2 0 h3
3 0.4 0.2 h3
2 0.6 0.4 h3
1 0 −0.2
−1 −0.5 1
0.8 0 4 0.5 1 −0.2
−1 1 4 h2 h3 0.8 0.6 6 h3 1 6 h4 h5 0.2 0 6 h2 0.5 0.4 0.2 6 0 0.6 0.4 −0.5 0 −0.2
−1 −0.5 0 0.5 1 −0.2
−1 −0.5 0 0.5 1 Figure 14.6: Plot of the Lagragian interpolants for diﬀerent polynomial degrees: linear
(top left), quadratic (top right), cubic (bottom left), and ﬁfth (bottom right). The
collocation points are shown by circles, and are located at the GaussLobatto roots of
the Legendre polynomial. The superscript indicates the total number of collocation
points, and the subscript the collocation point with which the polynomial interpolant is
associated. ξr1 ξr2 r r ξrm r ξ
r rP Figure 14.7: Distribution of collocation points in a spectral element. In this example
there are 8 collocation points (polynomial of degree 7). 228 CHAPTER 14. FINITE ELEMENT METHODS and are shown in ﬁgure 14.7. Equation 14.79 shows the two diﬀerent forms in we can
express the Lagragian interpolant; the traditional notation expresses hm as a product
of P − 1 factors chosen so as to guarantee the Lagragian property; the second form is
particular to the choice of Legendre GaussLobatto points Boyd (1989); Canuto et al.
(1988). It is easy to show that hm (ξn ) = δmn , and they are polynomials of degree P − 1.
Note that unlike the previous cases the collocation points are not equally spaced within
each element but tend to cluster more near the boundary of the element. Actually the
collocation spacing is O(1/(P − 1)2 ) near the boundary and O(1/(P − 1)) near the center
of the element. These functions are shown in ﬁgure 14.6 for P = 4 and 6. The P − 2
internal points are localized to a single element and do not interact with the interpolation
function of neighboring elements; the edge interpolation polynomials have support in two
neighboring elements.
The evaluation of the derivative of the solution at speciﬁed points ηn is equivalent to:
P u′ (ηn ) = h′ (ηn )um
m (14.81) m=1 and can be cast in the form of a matrix vector product, where the matrix entries are the
derivative of the interpolation polynomials at the speciﬁed points ηn .
The only problem arises from the more complicated form of the integration formula.
For this reason, it is common to evaluate the integrals numerically using high order
Gaussian quadrature; see section 14.2.10. Once the local matrices are computed the
assembly procedure can be performed with the local node numbering m running from 1
to P. 14.2.10 Numerical Integration Although it is possible to evaluate the integrals analytically for each interpolation polynomial, the task becomes complicated and error prone. Furthermore, the presence of
variable coeﬃcients in the equations may complicate the integrands and raise their order.
The problem becomes compounded in multidimensional problems. It is thus customary
to revert to numerical integration to ﬁll in the entries of the diﬀerent matrices.
Gauss quadrature estimates the deﬁnite integral of a function with the weighed sum
of the function evaluated at speciﬁed points called quadrature points:
Q 1 G
g(ξp )ωp + RQ g(ξ ) dξ =
−1 (14.82) p=1 G
where Q is the order of the quadrature formula and ξp are the Gauss quadrature points;
the superscript is meant to distinguish the quadrature points from the collocation points.
RQ is the remainder of approximating the integral with a weighed sum; it is usually
proportional to a high order derivative of the function g. Gauss Quadrature
The Gauss quadrature is one of the most common quadrature formula used. Its quadrature points are given by the roots of the Qth degree Legendre polynomial; its weights 14.2. FEM EXAMPLE IN 1D 229 and remainder are:
G
ωp = RQ = 2 (1 − ,
2
G
ξp )[L′ (ξp )]2
Q p = 1, 2, . . . , Q ∂ 2Q g
22Q+1 (Q!)4
3 ∂ξ 2Q
(2Q + 1)[(2Q)!] ξ (14.83) , ξ  < 1 (14.84) Gauss quadrature omits the end points of the interval ξ = ±1 and considers only interior
points. Notice that if the integrand is a polynomial of degree 2Q − 1; its 2Qderivative is
zero and the remainder vanishes identically. Thus a Q point Gauss quadrature integrates
all polynomials of degree less then 2Q exactly.
GaussLobatto Quadrature
The GaussLobatto quadrature formula include the end points in their estimate of the
integral. The roots, weights, and remainder of a Qorder GaussLobatto quadrature are:
GL
1 − ξp 2 GL
L′ −1 (ξp ) = 0
Q
GL
ωp
= RQ = (14.85) 2
, p = 1, 2, . . . , Q
2
(1 − ξp )[L′ (ξp )]2
Q −Q(Q − 1)3 22Q−1 [(Q − 2)!]4 ∂ 2Q−2 g
(2Q − 1)[(2Q − 2)!]3
∂ξ 2Q (14.86) ξ , ξ  <(14.87)
1 A Q point GaussLobatto quadrature of order Q integrates exactly polynomials of degree
less or equal to 2Q − 3.
Quadrature order and FEM
The most frequently encountered integrals are those associated with the mass matrix,
equation 14.67, and diﬀusion matrix, equation 14.66. We will illustrate how the order
of integration can be determined to estimate these integrals accurately. We assume for
the time being that the interpolation polynomial hm is of degree P − 1 (there are P
collocation points within each element), and the metric of the mapping is constant. The
integrand in equation 14.66 is of degree 2(P − 2), and the in equation 14.67 is of degree
2(P − 1).
If Gauss quadrature is used and exact integration is desired then the order of the
quadrature must be Q > P − 2 in order to evaluate 14.66 exactly and Q > P in order to
evaluate 14.67 exactly. Usually a single quadrature rule is chosen to eﬀect all integrations
needed. In this case Q = P + 1 will be suﬃcient to evaluate the matrices M and D in
equations 14.6714.66. Higher quadrature with Q > P + 1 may be required if additional
terms are included in the integrand; for example when the metric of the mapping is
not constant. Exact evaluation of the integrals using GaussLobatto quadrature requires
more points since it is less accurate then Gauss quadrature: Q ≥ (2P + 3)/2 GaussLobatto points are needed to compute the mass matrix exactly. 230 CHAPTER 14. FINITE ELEMENT METHODS Although the order of Gauss quadrature is higher, it is not always the optimal choice;
other considerations may favor GaussLobatto quadrature and reduced (inexact) integration. Consider a quadrature rule where the collocation and quadrature points are
identical, such a rule is possible if one chooses the GaussLobatto quadrature of order P ,
GL
where P is the number of points in each element; then ξm = ξm for m = 1, . . . , P . The
evaluation of the local mass matrix becomes:
j
Mm,n = ≈ 1
−1
P hm (ξ )hn (ξ ) ∂x
dξ
∂ξ hm (ξp )hm (ξp )
p=1
P = δm,p δn,p
p=1 = δm,n ∂x
∂ξ ∂x
∂ξ ωm ∂x
∂ξ
ωp (14.88)
ωp (14.89) ξp (14.90) ξp (14.91) ξm Equation 14.91 shows that mass matrix becomes diagonal when the quadrature and collocation points are identical. This rule applies equatlly well had we chosen the Gauss
points for quadrature and collocation. However, the GaussLobatto roots are preferred
since they simplify the imposition of C 0 continuity across elements (there are formulation
where C 0 continuity is not necessary, Gauss quadrature and collocation becomes sensible). The implication of a diagonal mass matrix is profound for it simpliﬁes considerably
the computations of timedependent problems. As we will see later, the timeintegration
requires the inversion of the mass matrix, and this task is inﬁnitely easier when the mass
matrix is diagonal. The process of reducing the mass matrix to diagonal is occasionally
referred to as mass lumping. One should be carefull when low order ﬁnite elements are
used to build the mass matrix as the reduced quadrature introduces error. For low order
interpolation (linear and quadratic) mass lumping reduces the accuracy of the ﬁnite element method substantially; the impact is less pronounced for higher order interpolation;
the rule of thumb is that mass lumping is terrible for linear element and has little impact
for P > 3.
Example 17 We solve the 1D problem: uxx − 4u = 0 in 0 ≤ x ≤ 1 subject to the boundary conditions u(0) = 0, and ux = 1. The analytical solution is u = sinh2x/(2cosh2).
The rms error between the ﬁnite element solution and the analytical solution is shown
in ﬁgures 14.8 as a function of the number of degrees of freedom. The plots show the
error for the linear (red) and quadratic (blue) interpolation. The left panel shows a
semi logarithmic plot to highlight the exponential convergence property of the spectral
element method, while the right panel shows a loglog plot of the same quantities to
show the algebraic decrease of the error as a function of resolution for the linear and
quadratic interpolation as evidenced by the straight convergence lines. The slopes of the
convergence curves for the spectral element method keeps increasing as the number of
degrees of freedom is increased. This decrease is most pronounced when the degrees of
freedom as added as increased spectral interpolation within each element as opposed to
increasing the number of elements. Note that the spectral element computations used 14.3. MATHEMATICAL RESULTS 231
−2 10 −2 10 −4 −4 1 10 1 10 2 2 −6 10 −6 10 3 3 4 4
5 −8 5 −8 10 ε2 ε2 10 −10 −10 10 −12 10 −14 10 10 −12 10 −14 10 −16 10 −16 10 0 50 100 150 200 250 300 350 0 1 10 2 10 3 10 10 K(N−1)+1 K(N−1)+1 Figure 14.8: Convergence curves for the ﬁnite element solution of uxx − 4u = 0. The
red and blue lines show the solutions obtained using linear and quadratic interpolation,
respectively, using exact integration; the black lines show the spectral element solution.
GaussLobatto quadrature to evaluate the integrals, whereas exact integration was used
for linear and quadratic ﬁnite elements. The inexact quadrature does not destroy the
spectral character of the method. 14.3 Mathematical Results The following mathematical results are presented without proof given the substantial
mathematical sophistication in their derivation. 14.3.1 Uniqueness and Existence of continuous solution The existence and uniqueness of the solution to the weak form is guaranteed by the
LaxMilgram theorem:
Theorem 1 LaxMilgram Let V be a Hilbert space with norm
V , consider the bilinear
form A(u, v ) : V × V −→ R, and the bounded linear functional F (v ) : V −→ R. If the
bilinear form is bounded and coercive,i.e. there exists positive constants ρ and β such
that
• continuity of A:
• coercivity of A: A(u, v ) ≤ β u
ρu V V v V ∀u, v ∈ V ≤ A(u, u) ∀u ∈ V Then there exists a unique u ∈ V such that
ˆ
A(ˆ, v ) = F (v ) ∀v ∈ V
u (14.92) The continuity condition guarantees that the operator A is bounded: A(u, v ) ≤
β u 2 V . This, in combination with the coercivity condition guarantees that A is norm
equivalent to V . The above theorem guarantees the existence and uniqueness of the
continuous solution. We take the issue of the discrete solution in the following section. 232 14.3.2 CHAPTER 14. FINITE ELEMENT METHODS Uniqueness and Existence of continuous solution The inﬁnitedimensional continuous solution u, must be approximated with a ﬁnite diˆ
mensional approximation uN where N characterizes the dimensionality (number of degrees of freedom) of the discrete solution. Let VN ⊂ V be a ﬁnite dimensional subspace
of V providing a dense coverage of V so that in the limit N → ∞, limN →∞ VN = V .
Since VN is a subset of V the condition of the LaxMilgram theorem are fullﬁlled and
hence a unique solution exists for the discrete problem. The case where the discrete space
VN ⊂ V (VN is a subset of V ) is called the conforming case. The proofs of existence
and uniqueness follow from the simple fact that VN is a subset of V . Additionally, for
the Galerkin approximation the following stability condition is satisﬁed:
uN V ≤C f (14.93) where C is a positive constant independent of N . One has moreover that:
un − u
ˆ V ≤ β
inf u − v V
ˆ
ρ v∈VN (14.94) The inequality (14.93) shows that the V norm of the numerical solution is bounded by
the L2 norm of the forcing function f (the data). This is essentially the stability criterion
of the LaxRichtmeyer theorem. Inequality (14.94) says that the V norm of the error is
bounded by the smallest error possible v ∈ VN to describe u ∈ V . By making further
ˆ
assumptions about the smoothness of the solution u it is possible to devise error estimates
ˆ
in terms of the size of the elements h. The above estimate guarantees that the left hand
side of (14.94) goes to zero as N → ∞ since VN → V . Hence the numerical solution
converges to the true solution. According to the LaxRichtmeyer equivalence theorem,
since two conditions (stability and convergence) hold, the third condition (consistency)
must follow; the discretization is hence also consistent. 14.3.3 Error estimates Inequality (14.94) provide the mean to bound the error in the numerical solution. Let
uI = IN u be the interpolant of u in VN . An upper bound on the approximation error
ˆ
ˆ
ˆ
can be obtained since
un − u
ˆ V ≤ β
inf v − u V ≤ C uI − u V
ˆ
ˆ
ˆ
ρ v∈VN (14.95) Let h reﬂect the charateristic size of an element (for a uniform discretization in 1D, this
would be equivalent to ∆x). One expects un − u V → 0 as h → 0. The rate at which
ˆ
that occurs depends on the smothness of the solution u.
ˆ
For linear intepolation, we have the following estimate:
1 un − u
ˆ H1 ˆ
≤ C h uH 2 , uH 2 =
ˆ 0 uxx dx 1
2 (14.96) where uH 2 is the socalled H 2 seminorm, (essentially a measure of the “size” of the
ˆ
second derivative), and C is a generic positive constant independent of h. If the solution 14.4. TWO DIMENSIONAL PROBLEMS 233 admits integrabale second derivatives, then the H 1 norm of the error decays linearly with
the grid size h. The L2 norm of the error however decreases quadratically according to:
un − u
ˆ H1 1 ˜
≤ C h uH 2 uH 2 =
ˆ
ˆ
2 0 2 (uxx ) dx 1
2 (14.97) The diﬀerence between the rate of convergences of the two error norms is due to the
fact that the H 1 norm takes the derivatives of the function into account. That the ﬁrst
derivative of u are approximated to ﬁrst order in h while u itself is approximated to
ˆ
ˆ
second order in h.
For an interpolation using polynomials of degree k the L2 norm of the error is given
by
ˆ
un − u ≤ C hk+1 uH k+1
ˆ
ˆ uH k+1 = ˆ 1
0 dk+1 u
dxk+1 2 1
2 dx (14.98) provided that the solution is smooth enough i.e. the k + 1 derivative of the solution is
squareintegrable.
For the spectral element approximation using a single element, the error depends on
N and the regularity of the solution. If u ∈ H m with m > 0, then the approximation
ˆ
error in the L2 norm is bounded by
ˆ
ˆ
un − u ≤ C N −m u Hm (14.99) The essential diﬀerence between the pversion of the ﬁnite element method and the
spectral element method lies in the exponent of the error (note that h N −1 ). In the
pcase the exponent of N is limited by the degree of the interpolating polynomial. In
the spectral element case it is limited by the smoothness of the solution. If the latter is
inﬁnitely smooth then m ≈ N and hence the decay is exponential N −N . 14.4 Two Dimensional Problems The extension of the ﬁnite element methods to twodimensional elliptic problems is
straightforward and follows the same lines as for the onedimensional examples, namely:
transformation of the PDE to a variational problem, restriction of this problem to a
ﬁnite dimensional space (eﬀectively, the Galerkin step), discretization of the geometry
and spatial interpolation, assembly of the stiﬀness matrix, imposition of boundary conditions, and ﬁnally solution of the linear system of equations. Two dimensional problems
have more complicated geometries then onedimensional problems. The issue of geometry discretization becomes important and this will be explored in the present section.
We will take as our sample PDE the following problem:
∇2 u − λu + f = 0, x ∈ Ω (14.100) ∇u · n = q, x ∈ ΓN (14.102) u(x) = ub (x), x ∈ ΓD (14.101) 234 CHAPTER 14. FINITE ELEMENT METHODS where Ω is the region of interest with ∂ Ω being its boundary, ΓD is that portion of
∂ Ω where Dirichlet conditions are imposed and ΓN are those portions where Neumann
conditions are imposed. We suppose the ΓD + ΓN = ∂ Ω.
The variational statements comes from multiplying the PDE by test functions v (x)
that satisfy v (x ∈ ΓD ) = 0, integrating over the domain Ω, and applying the boundary
conditions; the variational statement boils down to:
Ω (∇u · ∇v + λuv ) dA = f v dA +
Ω ΓN 1
vq ds, ∀v ∈ H0 (14.103) 1
where H0 is the space of square integrable functions on Ω that satisfy homogenous
boundary conditions on ΓD ; ds is the arclength along the boundary. The Galerkin
formulation comes from restricting the test functions to a ﬁnite set and interpolation
functions to a ﬁnite set:
N u= ui φi (x), v = φj , j = 1, 2, . . . , N (14.104) i=1 where the φi (x) are now two dimensional interpolation functions (here we restrict ourselves to Lagrangian interpolants). The Galerkin formulations becomes ﬁnd ui such that
N
i=1 Ω (∇φi · ∇φi + λφi φj ) dAui = Ω f φj dA + ΓN φj q ds, j = 1, 2, . . . , N (14.105) We thus recover the matrix formulation Ku = b where the stiﬀness matrix and forcing
functions are given by:
Kji =
bj Ω =
Ω (∇φi · ∇φi + λφi φj ) dA (14.106) f φj dA + (14.107) ΓN φj q ds In two space dimensions we have a greater choice of element topology (shape) then in
the simplistic 1D case. Triangular elements, the simplex elements of 2D space, are very
common since they have great ﬂexibility, and allow the discretization of very complicated
domains. In addition to triangular elements, quadrilaterals elements with either straight
or curved edges are extensively used. In the following, we explore the interpolation
functions for each of these elements, and the practical issues needed to be overcome in
order to compute the integrals of the Galerkin formulation. 14.4.1 Linear Triangular Elements One of the necessary ingredient of FE computations is the localization of the computations to an element which requires the development of a natural coordinate system in
which to perform the computations. For the triangle, the natural coordinate system is
the area coordinate shown in ﬁgure 14.9. The element is identiﬁed by the three nodes
i, j and k forming its vertices; let P be a point inside the triangle with coordinates x.
By connecting P to the three vertices i, j , and k, we can divide the elements into three 14.4. TWO DIMENSIONAL PROBLEMS 235 k h hh
h
hhhh
d
hhh
d
hh
2
2
Ai
d
222 j
22
d22
c
P
Aj
Ak
i
Figure 14.9: Natural area coordinate system for triangular elements.
small triangles with areas Ai , Aj and Ak , respectively with Ai + Aj + Ak = A, where
A is the area of the original element. Notice that if the point P is located along edge
j − k, Ai = 0, whereas if its located at i we have Ai = A and Aj = Ak = 0; the nodes j
and k have similar property. This natural division of the element into 3 parts of similar
structure allows us to deﬁne the area coordinate of point P as:
ai = Ai
Aj
Ak
, aj =
, ak =
, ai + aj + ak = 1
A
A
A (14.108) The area of a triangle is given by the determinant of the following matrix:
1
A=
2 1 xi yi
1 xj y j
1 xk yk (14.109) The other areas Ai , Aj and Ak can be obtained similarly; their dependence on the
coordinate (x, y ) of the point P is linear. It is now easy to verify that if we set the
local interpolation functions to φi = ai we obtain the linear Lagrangian interpolant
on the triangle, i.e. that φi (xm ) = δi,m where m can tak the value i, j , or k. The
linear Lagrangian interpolant for point i can be easily expressed in terms of the global
coordinate system (x, y ):
1
(αi + βi x + γi y )
2A
= xj yk − xk yj φi (x, y ) = (14.110) αi (14.111) βi = yj − yk γi = −(xj − xk ) (14.112)
(14.113) The other interpolation function can be obtained with a simple permutation of indices.
Note that the derivative of the linear interpolation functions are constant over the element. The linear interpolation now takes the simple form:
u(x) = ui φi (x) + uj φj (x) + uk φk (x) (14.114) 236 CHAPTER 14. FINITE ELEMENT METHODS 1
1
0.5
0.5 1 0
1 0.5
0
0 0.2 0.4 0.6 0.8 1 1 0.5 0.5 0 0 0 1 0.5 0
1
1 0.5 0.5
0 0 Figure 14.10: Linear interpolation function over linear triangular elements. The triangle
is shown in dark shades. The three interpolation functions are shown in a light shade
and appear as inclined planes in the plot.
where ui , uj and uk are the values of the solution at the nodes i, j and k. The interpolation formula for the triangle guarantees the continuity of the solution across element
boundaries. Indeed, on edge j − k for example, the interpolation does not involve node
i and is essentially a linear combination using the functions values at node j and k, thus
ensuring continuity. The linear interpolation function for the triangle are shown in ﬁgure
14.10 as a threedimensional plot.
The usefullness of the area coordinates stems from the existence of the following
integration formula over the triangle:
p!q !r !
(14.115)
ap aq ar dA = 2A
ijk
(a + b + c + 2)!
A
where the notation p! = 12 . . . p stands for the factorial of the integer p. It is now easy
to verify that the local mass matrix is now given by φi φi φi φj Me = φj φi φj φj
A
φk φi φk φj 211
φi φk
A φj φk dA =
1 2 1
12
112
φk φk (14.116) The entries of the matrix arising from the discretization of the Laplace operator are
easy to compute since the gradients of the interpolation and test functions are constant
over an element; thus we have:
De = ∇φi · ∇φi ∇φi · ∇φj ∇φj · ∇φi ∇φj · ∇φj
A
∇φk · ∇φi ∇φk · ∇φj ∇φi · ∇φk ∇φj · ∇φk dA
∇φk · ∇φk (14.117) 14.4. TWO DIMENSIONAL PROBLEMS 237 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 14.11: FEM grid and contours of the solution to the Laplace equation in a circular
annulus. = β β + γi γi βi βj + γi γj βi βk + γi γk
1 ii βj βi + γj γi βj βj + γj γj βj βk + γj γk 4A
βk βi + γk γi βk βj + γk γj βk βk + γk γk (14.118) As an example of the application of the FEM element method we solve the Laplace
equation in a circular annulus subject to Dirichlet boundary conditions using a FEM with
linear triangular elements. The FEM grid and contours of the solution are shown in ﬁgure
14.11. The grid contains 754 nodes and 1364 elements. The boundary conditions were
set to cos θ on the outer radius and sin θ on the inner radius, where θ is the azimuthal
angle. The inner and outer radii are 1/2 and 1, respectively. The contour lines were
obtained by interpolating the FEM solution to a high resolution (401x401) strutured
grid prior to contouring it. 14.4.2 Higher order triangular elements It is possible to deﬁne higher order interpolation formula within an element without
changing the shape of an element. For example the quadratic triangular elements with
collocation points at the triangle vertices and midpoints of edges are given by
u(x) = ui φ2 (x) + uj φ2 (x) + uk φ2 (x)
j
k
i
φ2
i
φ2
j
2
φk
2
φi−j + ui−j φ2−j (x) + uj −k φ2−k (x) + uk−i φ2 −i (x)
i
j
k
= ai (ai − 1) = aj (aj − 1) = ak (ak − 1)
= 4ai aj (14.119)
(14.120)
(14.121)
(14.122)
(14.123) 238 CHAPTER 14. FINITE ELEMENT METHODS
η
3u T 1 E 1
1 1
u 2
u
4
4
u
ξ u 1 2 y T u
g
g
g
g
u
&1
&
&
&
g&
3 gu
&
Ex Figure 14.12: Mapping of a quadrilateral between the unit square in computational space
(left) and physical space (right).
φ2−k = 4aj ak
j (14.124) φ2 −i = 4ak ai
k (14.125) where i − k denotes the midpoint of edge i − k. There are 6 degrees of freedom associated
with each quadratic triangular element. Although it is possible to deﬁne even higher order
interpolation in the triangle by proceeding as before; it is not so simple to implement.
The alternative is to use a mapping to a structured computational space and deﬁne
the interpolation and collocation function in that space. It is important to choose the
collocation points appropriately in order to avoid Gibbs oscillations as the polynomial
degree increases. Spectral triangular elements are the optimal choice with this regard.
We will not discuss spectral element triangles here; we refer the interested reader to
Karniadakis and Sherwin (1999). 14.4.3 Quadrilateral elements Change of Coordinates
The derivation of the interpolation and integration formula for quadrilateral ﬁnite elements follows the line of the previous section. The main task resides in deﬁning the
local coordinate system in the master element shown in ﬁgure 14.12. For straight edged
elements the mapping between physical and computational space can be easily eﬀected
through the following bilinear map:
x(ξ, η ) = 1−ξ
2 1+ξ
1+η
1−η
x1 +
x3 +
2
2
2 1+η
1−η
x2 +
x4
2
2 (14.126) In order to derive all the expressions needed to express the Galerkin integrals in
computational space, it is useful to introduce a basis in computational space (eξ , eη )
tangential to the coordinate lines (ξ, η ); we denote by (x, y ) the coordinate in physical
space. Let r = xi + y j denotes a vector pointing to point P located inside the element.
We deﬁne the basis vectors vectors tangent to the coordinate lines as
eξ = ∂r
∂r
= xξ i + y ξ j , e η =
= xη i + y η j
∂ξ
∂η (14.127) 14.4. TWO DIMENSIONAL PROBLEMS 239 where r denotes the position vector of a point P in space and (i, j) forms an othonormal
basis in the physical space. Inverting the above relationship one obtains
i= yξ
xη
xξ
yη
eξ − eη , j = − eξ + eη
J
J
J
J where J = xξ yη − xη yξ is the Jacobian of the mapping. The norms of eξ and eη are given
by
eξ 2 = eξ · eξ = (xξ )2 + (yξ )2 , eη 2 = eη · eη = (xη )2 + (yη )2
The basis in the computational plane is orthogonal if eξ · eη = xξ xη + yξ yη = 0; in general
the basis is not orthogonal unless the element is rectangular.
It is now possible to derive expression for length and area segements in computational
space. These are needed in order to compute boundary and area integrals arising from
the Galerkin formulation. Using the deﬁnition ( ds)2 = dr · dr with dr = rξ dξ + rη dη ,
we have:
( ds)2 = eξ dξ + eη dη 2 = eξ 2 dξ 2 + eη 2 dη 2 + 2eξ · eη dξ dη (14.128) The diﬀerential area of a curved surface is deﬁned as the area of its tangent plane
approximation (in 2D, the area is always ﬂat.) The area of the parallelogram deﬁned by
the vectors dξ eξ and dη eη is dA =  dξ eξ × dη eη  = i
jk
xξ yξ 0
xη yη 0 dξ dη = xξ yη − xη yξ  dξ dη = J  dξ dη (14.129)
after using the deﬁnition of (eξ , eη ) in terms of (i, j).
Since x = x(ξ, η ) and y = y (ξ, η ), the derivative in physical space can be expressed
in terms of derivatives in computational space by using the chain rule of diﬀerentiation;
in matrix form this can be expressed as:
ux
uy = ξx ηx
ξy ηy uξ
uη (14.130) Notice that the chain rule involves the derivatives of ξ, η in terms of x, y whereas the
bilinear map readily delivers the derivatives of x, y with respect to ξ, η . In order to avoid
inverting the mapping from physical to computational space we derive expressions for
∇ξ , and ∇η in terms of xξ , xη , etc... Applying the chain rule to x and y we obtain
(noticing that the two variables are independent) the system of equations:
xξ yξ
xη yη ξx ηx
ξy ηy = xx y x
xy yy = 10
01 (14.131) The solution is
ξx ηx
ξy ηy = 1
J y η −y ξ
−xη
xξ , J = xξ yη − xη yξ (14.132) 240 CHAPTER 14. FINITE ELEMENT METHODS 1 1 0.5 0.5 0
1 0
1
1 0
−1 1 0 0 0
−1 −1 1 1 0.5 0.5 0
1 −1 0
1
1 0
−1 1 0 0 0
−1 −1 −1 Figure 14.13: Bilinear shape functions in quadrilateral elements, the upper left hand
corner is h1 (ξ )h1 (η ), the upper right hand side panel shows h2 (ξ )h1 (η ), the lower left
panel shows h1 (ξ )h2 (η ) and lower right shows h2 (ξ )h1 (η ).
For the bilinear map of equation 14.126, the metrics and Jacobian can be easily
computed by diﬀerentiation:
1 − η x2 − x1 1 + η x4 − x1
+
2
2
2
2
1 − ξ x3 − x1 1 + ξ x4 − x2
xη =
+
2
2
2
2 xξ = 1 − η y2 − y1 1 + η y4 − y1
+
(14.133)
2
2
2
2
1 − ξ y3 − y1 1 + ξ y4 − y2
yη =
+
(14.134)
2
2
2
2
yξ = The remaining expression can be obtained simply by plugging in the the various expressions derived earliear. 14.4.4 Interpolation in quadrilateral elements The interpolation of the solution within quadrilateral elements is easily accomplished
if tensorized product of onedimensional Lagrangian interpolants are used to build the
twodimensional formula. For the bilinear map shown in ﬁgure 14.12, for example we 14.4. TWO DIMENSIONAL PROBLEMS 241 can use the collocation points located at the vertices of the quadrilateral to deﬁne the
following interpolation:
4 u(ξ, η ) = um φm (ξ, η ) = u1 φ1 + u2 φ2 + u3 φ3 + u4 φ4 (14.135) m=1 where the twodimensional Lagrangian interpolants are tensorized product of the onedimensional interpolants deﬁned in equation 14.61:
φ1 (ξ, η ) = h1 (ξ )h1 (η ), φ2 (ξ, η ) = h2 (ξ )h1 (η ),
φ3 (ξ, η ) = h1 (ξ )h2 (η ), φ4 (ξ, η ) = h2 (ξ )h2 (η ), (14.136) and shown in ﬁgure 14.13. Note that the bilinear interpolation function above satisfy
the C 0 continuity requirement. This can be easily veriﬁed by noting ﬁrst that the interpolation along an edge involves only the collocation points along that edge, hence
neighboring elements sharing an edge will interpolate the solution identically if the value
of the function on the collocation points is unique. Another important feature of the
bilinear interpolation is that, unlike the linear interpolation in triangular element, it
contains the term of second degree: ξη . Hence the interpolation within an element is
nonlinear; it is only linear along edges.
Before proceeding further, we introduce a new notation for interpolation with quadrilaterals to explicitely bring out the tensorized form of the interpolation functions. This
is accomplished by breaking the single twodimensional index m in equation 14.135 into
two onedimensional indices (i, j ) such that m = (j − 1)2 + i, where (i, j ) = 1, 2. The
index i runs along the ξ direction and the index j along the η direction; thus m = 1
becomes identiﬁed with (i, j ) = (1, 1), m = 2 with (i, j ) = (2, 1), etc... The interpolation
formula can now be written as
2 2 u(ξ, η ) = uij hi (ξ )hj (η ) (14.137) j =1 i=1 where uij are the function values at point (i, j ).
With this notation in hand, it is now possible to write down arbitrarily high order Lagrangian interpolation in 2D using the tensor product formula. Thus, a 1D interpolation
using N points per 1D element can be extended to 2D via:
N N uij hN (ξ )hN (η )
i
j u(ξ, η ) = (14.138) j =1 i=1 The superscript N has been introduced on the Lagragian interpolants to stress that
they are polynomials of degree N − 1 and use N collocation points per direction. The
collocation points using 7th degree interpolation polynomials is shown in ﬁgure 14.14.
Note, ﬁnally, that sum factorization algorithms must be used to compute various
quantities on structured sets of points p, q in order to reduce the computational overhead.
For example, the derivative of the function at point (p, q ), can be computed as:
N uξ ξp ,ηq = j =1 N uij
i=1 dhN
i
dξ ξp hN (ηq ).
j (14.139) 242 CHAPTER 14. FINITE ELEMENT METHODS Figure 14.14: Collocation points within a spectral element using 8 collocation points per
direction to interpolate the solution; there are 82 =64 points in total per element.
First the term in parenthesis is computed and saved in a temporary array, second, the ﬁnal expression is computed and saved; this essentially reduces the operation from O(N 4 )
to O(N 3 ). Further reduction in operation count can be obtained under special circumstances. For instance, if ηq happens to be a collocation point, then hN (ηq ) = δjq and the
j
formula reduces to a single sum:
N uξ ξp ,ηq = 14.4.5 uiq
i=1 dhN
i
dξ (14.140)
ξp Evaluation of integrals The integrals needed to build the stiﬀness matrix for two dimensional quadrilateral elements are a bit more complicated then those encountered in triangular elements. This
is primarily due to the lack of magic integration formula, and the more complex (nonconstant) mapping between physical and computational space. We start by considering
the calculation of the elemental mass matrix Mm,n = A φm φn dA; in computational
space and using the 2index notation introduced earlier, this integral becomes:
e
Mij,kl = 1 1 −1 −1 hi (ξ ) hj (η ) hk (ξ ) hl (η ) J  dξ dη (14.141) where m = (j − 1)P + i, and n = (k − 1)P + l. In order to bypass the tediousness
of evaluating integrals analytically for each term that may be present in the Galerkin
formulation, it is common practice to evaluate the integrals with Gauss quadrature.
The order of the quadrature needed depends of course on the polynomial degree of the
integrand and on whether exact integration is required.
We now proceed to determine the quadrature order needed to integrate the mass
matrix exactly. In the case of the bilinear map of equation 14.126, the Jacobian varies
bilinearly over the element, hence it is linear in the variable ξ and η . Assuming the
Lagrange interpolation uses P points in each direction, the integrand in the mass matrix
is a polynomial of degree 2(P − 1) + 1 in each of the variables ξ and η . Thus, Gauss 14.4. TWO DIMENSIONAL PROBLEMS 243 quadrature of order Q will evaluate the integrals exactly provided Q ≥ P :
Q Q Mij,kl =
m=1 n=1 G
G
G
G
GG
hi (ξm ) hj (ηn ) hk (ξm ) hl (ηn ) Jmn ωm ωn (14.142) GG
G
where Jmn denotes the Jacobian at the Gauss quadrature points (ξm , ηn ), and ωm are
the Gauss quadrature weights. The only required operations are hence the evaluation of
G
the Lagrangian interpolants at the points ξm , and summations of the terms in 14.142. If
GaussLobatto integration is used, the roots and weights become those appropriate for
the GaussLobatto quadrature; the number of quadrature points needed to evaluate the
integral exactly increases to Q ≥ P + 1.
Like the onedimensional case, the mass matrix can be rendered diagonal if inexact
(but accurate enough) integration is acceptable. If the quadrature points and collocation
G
points coincide, hk (ξm ) = δim , and the expression for the mass matrix entries reduces to: Mij,kl = δik δjl ωk ωl Jkl  (14.143) The integrals involved in evaluating the discrete Laplace operator, the matrix D ,
is substantially more complicated in the present instance. Here we continue with our
reliance on Gauss quadrature and derive the terms that must be computed. Keeping
with our 2 index notation, we have
Dij,kl = A ∇φij · ∇φkl dA. (14.144) Most of the work comes from having to express the inner product in the integrand in
computational space:
∇φij · ∇φkl =
=
=
+ ∂φij ∂φkl ∂φij ∂φkl
+
∂x ∂x
∂y ∂y
∂ φkl
∂φij
∂φkl
∂ φij
∇ξ +
∇η ·
∇ξ +
∇η
∂ξ
∂η
∂ξ
∂η
∂φij ∂φkl
∂φij ∂φkl
∇ξ · ∇ξ +
∇η · ∇η
∂ξ ∂ξ
∂η ∂η
∂ φij ∂φkl ∂φij ∂φkl
+
∇ξ · ∇η.
∂ξ ∂η
∂η ∂ξ (14.145)
(14.146) (14.147) Setting φij = hi (ξ )hj (η ) and φkl = hk (ξ )hl (η ), and evaluating the integrals using Gauss
quadrature, we get:
Q Q Dij,kl =
n=1 m=1
Q
Q +
n=1 m=1
Q
Q +
n=1 m=1
Q
Q +
n=1 m=1 h′ (ξm ) hj (ηn ) h′ (ξm ) hl (ηn ) [∇ξ · ∇ξ J ]m,n ωm ωn
i
k
hi (ξm ) h′ (ηn ) hk (ξm ) h′ (ηn ) [∇η · ∇η J ]m,n ωm ωn
j
l
h′ (ξm ) hj (ηn ) hk (ξm ) h′ (ηn ) [∇ξ · ∇η J ]m,n ωm ωn
i
l
hi (ξm ) h′ (ηn ) h′ (ξm ) hl (ηn ) [∇ξ · ∇η J ]m,n ωm ωn (14.148)
j
k 244 CHAPTER 14. FINITE ELEMENT METHODS where the expressions in bracket are evaluated at the quadrature points (ξm , ηn ) ( we
have omitted the superscript G from the quadrature points). Again, substantial savings
can be achieved if the GaussLobatto quadrature of the same order as the inteporlation
polynomial is used. The expression for Dij,kl reduces to:
Q Dij,kl = δjl
m=1
Q + δik
n=1 h′ (ξm ) h′ (ξm ) [∇ξ · ∇ξ J ]m,j ωm ωj
i
k
h′ (ηn ) h′ (ηn ) [∇η · ∇η J ]i,n ωi ωn
j
l + h′ (ξk ) h′ (ηj ) [∇ξ · ∇η J ]k,j ωk ωj
i
l + h′ (ξi ) h′ (ηl ) [∇ξ · ∇η J ]i,l ωi ωl
k
j 14.5 (14.149) Timedependent problem in 1D: the Advection Equation Assume we attempt to solve the 1D constantcoeﬃcient advection equation:
ut + cux = 0, x ∈ Ω (14.150) subject to appropriate initial and boundary conditions. The variational statement that
solves the above problem is: Find u such that
Ω (ut + cux )vdx = 0, ∀v (14.151) The Galerkin formulation reduces the problem to a ﬁnite dimensional space by replacing
the solution with a ﬁnite expansion of the form
N u= ui (t)φi (x) (14.152) i=1 and setting the test functions to v = φj , j = 1, 2, . . . , N . Note that unlike the steady
state problems encountered earlier ui , the value of the function at the collocation points,
depends on time. Replacing the ﬁnite expansion in the variational form we get the
following ordinary diﬀerential equation (ODE)
M du
+ Cu = 0
dt (14.153) where u is the vector of solution values at the collocation points, M is the mass matrix,
and C the matrix resulting from discretization of the advection term using the Galerkin
procedure. The entries of M and C are given by:
L Mj,i = φi φj dx 0
L Cj,i = c
0 ∂φi
φj dx
∂x (14.154)
(14.155) 14.5. TIMEDEPENDENT PROBLEM IN 1D: THE ADVECTION EQUATION 245
We follow the procedure outlined in section 14.2.7 to build the matrices M and C .
We also start by looking at linear interpolation functions as those deﬁned in equation
14.60. The local mass matrix is given in equation 14.68; here we derive expressions for
the advection matrix C assuming the advective velocity c is constant. and advection
matrices are
xj Cji = x j −1 hj dhi
dx = c
dx 1
−1 hi (ξ ) dhj (ξ )
c
dξ, C =
dξ
2 −1 1
−1 1 (14.156) After stiﬀness assembly, the global system of equations becomnes:
∆x d
c
(uj −1 + 4uj + uj +1 ) + (−uj −1 + uj +1 ) = 0
6 dt
2 (14.157) The approximation of the advective term using linear ﬁnite element has resulted in a
centereddiﬀerence approximation for that term; whereas the timederivative term has
produced the mass matrix. Notice that any integration scheme, even an explicit ones
like leapfrog, would necessarily require the inversion of the mass matrix. Thus, one can
already anticipate that the computational cost of solving the advection using FEM will
be higher then a similarly conﬁgured explicit ﬁnite diﬀerence method. For linear elements
in 1D, the mass matrix is tridiagonal and the increase in cost is minimal since tridiagonal
solvers are very eﬃcient. Quadratic elements in 1D lead to a pentadiagonal system and
is hence costlier to solve. This increased cost may be justiﬁable if it is compensated by
a suﬃcient increase in accuracy. In multidimensions, the mass matrix is not tridiagonal
but has only limited bandwidth that depends on the global numbering of the nodes; thus
even linear elements would require a full matrix inversion.
Many solutions have been proposed to tackle the extracost of the full matrix. One
solution is to use reduced integration using GaussLobatto quadrature which as we saw in
the Gaussian quadrature section leads immediately to a diagonal matrix; this procedure
is often referred to as mass lumping. For low order elements, mass lumping degrades
signiﬁcantly the accuracy of the ﬁnite element method, particularly in regards to its phase
properties. For the 1D advection equation mass lumping of linear elements is equivalent
to a centered diﬀerence approximation. For high order interpolation, i.e. higher then
degree 3, the loss of accuracy due to the inexact quadrature is tolerable, and is of the
same order of accuracy as the interpolation formula. Another alternative revolves around
the use of discontinuous test functions and is appropriate for the solution of mostly
hyperbolic equations; this approach is dubbed the Discontinuous Galerkin method, and
will be examined in a following section.
The system of ODE can now be integrated using one of the timestepping algorithms,
for example second order leapfrog, third order AdamsBashforth, or one of the RungeKutta schemes. For linear ﬁnite elements, the stability limit can be easily studied with
the help of Von Neumann stability analysis. For example, it is easy to show that a leapfrog scheme applied to equation will result in a stability limit of the form µ = c∆t/∆x <
√
1/ 3; and hence is much more restrictive then the ﬁnite diﬀerence scheme which merely
requires that the Courant number be less then 1. However, an examination of the phase
property of the linear FE scheme will reveal its superiority over centered diﬀerences. 246 CHAPTER 14. FINITE ELEMENT METHODS
Numerical phase speed of various spatial discretization Group velocity 1 1 0.9
0.5 0.8
FE1 0 0.6 CD6 −0.5 0.5 CD4 cg num /c an 0.7 CD2
CD4 −2 CD6 c −1 −1.5 0.4
CD2 0.3
0.2 −2.5 0.1
0
0 0.2 0.4 k ∆ x/π 0.6 0.8 1 −3
0 FE1
0.2 0.4 k∆ x/π 0.6 0.8 1 Figure 14.15: Comparison of the dispersive properties of linear ﬁnite elements with
centered diﬀerences, the left panel shows the ratio of numerical phase speed to analytical
phase speed, and the right panel shows the ratio of the group velocity
We study the phase properties implied by the linear ﬁnite element discretization by
looking for the periodic solution, in space and time, of the system of equation 14.157:
u(x, t) = ei(kxj −σt) . We thus get the following dispersion relationship and phase speed:
σ=
cF
c = 3c sin k∆x
∆x 2 + cos k∆x
3 sin k∆x
k∆x(2 + cos k∆x) (14.158)
(14.159) The numerical phase speed should be contrasted to the one obtaiend from centered second
and fourth order ﬁnite diﬀerences:
cCD2
c
cCD4
c =
= sin k∆x
k ∆x
1
sin k∆x sin 2k∆x
−
4
3
k ∆x
2k∆x (14.160)
(14.161) Figure 14.15 compares the dispersion of linear ﬁnite element with that of centered difference schemes of 2, 4 and 6 order. It is immediately apparent that the FE formulation
yields more accurate phase speed at all wavenumbers, and that the linear interpolation
is equivalent, if not slightly better then, a sixthorder centered FD approximation; in
particular the intermediate to short waves travel slightly faster then in FD. The group
velocity, shown in the right panel of ﬁgure 14.15, shows similar results for the long to
intermediate waves. The group velocity of the short wave are, however, in serious errors
for the FE formulation; in particular they have a negative phase speed and propagate upstream of the signal at a faster speed then the ﬁnite diﬀerence schemes. A masslumped
version of the FE method would collapse the FE curve onto that of the centered second
order method. 14.5. TIMEDEPENDENT PROBLEM IN 1D: THE ADVECTION EQUATION 247 14.5.1 Numerical Example As an example of the application of the FEM to the the inviscid 1D advection equation
we solve the following problem:
12 ut + ux = 0 on − 1 ≤ x ≤ 1, u(−1, t) = 0, u(x, 0) = e256(x+ 2 ) (14.162) The initial condition consists of an inﬁnitely smooth Gaussian hill centered at x = −1/2.
The solution at time t = 1 should be the same Gaussian hill but centered at x = 1/2. The
FEM solutions are shown in ﬁgure 14.16 where the consistent mass (exact integration)
and “lumped” mass solutions are compared, for various interpolation orders; the number
of elements was kept ﬁxed and the convergence study can be considered an preﬁnement
strategy. The number of element was chosen so that the hill is barely resolved for
linear elements (the case m = 2). It is obvious that the solution improves rapidly as m
(the number of interpolation points within an element) is increased. The lumped mass
solution is very poor indeed for the linear case where its poor dispersive properties have
generated substantial oscillatory behavior. The situation improves substantially when
the interpolation is increased to quadratic; the biggest discrepancy occuring near the
upstream foot of the hill. The lumped and consistent mass solutions are indistinguishable
for m = 5.
One may object that the improvements are solely due to the increased resolution.
We have repeated the experiment above trying to keep the total number of degrees of
freedom ﬁxed; in this case we are trying to isolate the eﬀects of improved interpolation
order. The ﬁrst case considered was an underresolved case using 61 total degrees of
freedom, and the comparison is shown in ﬁgure 14.17. The ﬁgure shows indeed the
improvements in the dispersive characteristics of the lumped mass approximation as the
degree of the polynomial is increased. The consistent and lumped solutions overlap over
the main portion of the signal for m ≥ 4 but diﬀer over the (numerically) dispersed waves
trailing the hill. These are evidence that the two approximations are still underresolved.
The solution in the resolved case is shown in ﬁgure 14.18 where the total number of
degrees of freedom is increased to 121. In this case the two solution overlap for m ≥ 3.
The two solutions are indistinguishable for m = 5 over the entire domain, evidence that
the solution is now wellresolved. Notice that the dispersive error of the lumped mass in
the linear interpolation case is still quite in error and further increase in the number of
elements is required; these errors are entirely due to masslumping as the consitent mass
solution is free of dispersive ripples.
The lesson of the previous example is that the dispersive properties of the lumpedmass solution is quite good provided enough resolution is used. The number of elements
needed to reach the resolution threshold decreases dramatically as the polynomial order
is increased. The dispersive properties of the lumpedmass linear interpolation FEM
seem to be the worse, and seem to require a verywell resolved solution before the eﬀect
of the masslumping is eliminated. One then has to weigh the increased cost of a masslumped large calculation versus that of a smaller consistentmass calculation to decided
whether lumping is costeﬀective or not. 248 CHAPTER 14. FINITE ELEMENT METHODS 1
0.5
0
−1 m=2
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1
0.5
0
−1 m=3
−0.8 1
0.5
0
−1 m=4
−0.8 1
0.5
0
−1 m=5
−0.8 Figure 14.16: Solution of the advection equation with FEM. The black lines show the result of the consistent mass matrix calculation and the red lines show that of the “lumped”
mass matrix calculation. The discretization consisted of 40 equally spaced elements on
the interval [0 1], and a linear (m=2), quadratic (m=3), cubic (m=4) and quartic (m=5)
interpolation. The time stepping is done with a TVDRK3 scheme (no explicit dissipation
included). 14.5. TIMEDEPENDENT PROBLEM IN 1D: THE ADVECTION EQUATION 249 1
0.5
0
−1
1
0.5
0
−1
1
0.5
0
−1
1
0.5
0
−1
1
0.5
0
−1
1
0.5
0
−1 m=2, E=60
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 m=3, E=30
−0.8 −0.6 m=4, E=20
−0.8 −0.6 m=5, E=15
−0.8 −0.6 m=6, E=12
−0.8 −0.6 m=7, E=10
−0.8 −0.6 Figure 14.17: Consistent versus lumped solution of the advection equation with FEM
for a coarse resolution with the total number of degrees of freedom ﬁxed, N ≈ 61. The
black and red lines refer to the consistent and lumped mass solutions, respectively. 250 CHAPTER 14. FINITE ELEMENT METHODS 1
0.5
0
−1 m=2, E=120
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1
0.5
0
−1 m=3, E=60
−0.8 −0.6 1
0.5
0
−1 m=4, E=40
−0.8 −0.6 1
0.5
0
−1 m=5, E=30
−0.8 −0.6 Figure 14.18: Consistent versus lumped solution of the advection equation with FEM
for a coarse resolution with the total number of degrees of freedom ﬁxed, N ≈ 121. The
black and red lines refer to the consistent and lumped mass solutions, respectively. 14.6. THE DISCONTINUOUS GALERKIN METHOD (DGM) 14.6 251 The Discontinuous Galerkin Method (DGM) A major drawback of the continuous Galerkin method is the requirement to maintain C 0
continuity; a requirement that leads to a tight coupling between neighboring elements.
In particular, it causes the mass matrix to be global which necessitates matrix inversion
to timestep the solution. There are classes of problem where the C 0 continuity is not
necessary; test and trial functions can then be made discontinuous, and the solution
process becomes considerably more cost eﬀective. This new formulation of the Galerkin
method has been dubbed the Discontinuous Galerkin Method (DGM). It is most suitable
to compute the solution to problems governed primarily by hyperbolic (like the pure
advection equation or the shallow water equation) or predominantly hyperbolic (like the
advectiondiﬀusion equation with high Peclet number).
In the following we describe the formulation of DGM for the simple case of a pure
advection equation:
Tt + v · ∇T = 0
(14.163)
where v is the advective velocity ﬁeld. If v is divergencefree, that is ∇ · v = 0, the
advection equation can be written in the conservative form
Tt + ·∇F = 0, F = vT (14.164) where F is the ﬂux and is a function of T ; equation 14.164 is written in the form of a
conservation law. We suppose that the domain of interest has been divided into elements.
Then the following variational statement applies inside each element: E (Tt + ·∇F)w dV = 0 (14.165) where w are the test functions, and E is the element of interest. If w is suﬃciently smooth,
we can integrate the divergence term by part using the Gauss theorem to obtain: E (Tt w − F · ∇w) dV + ∂E w F · n dS = 0 (14.166) where ∂E is the boundary of element E , n is the unit outward normal. The boundary
integral represent a weighed sum of the ﬂuxes leaving/entering the element. The discretization steps consist of replacing the inﬁnite dimensional space of the test functions
by a ﬁnite space, and representing the solution by a ﬁnite expansion Th . Since Th is discontinuous at the edges of elements, we must also replace the ﬂux F(T ) by a numerical
ﬂux G that depends on the values of Th outside and inside the element:
G = G(T i , T o ) (14.167) where T i and T o represent the values of the function on the edge as seen from inside
element, and from the neighboring element, respectively. Physical intuition dictates that
the right ﬂux is that obtained by following the characteristic (Lagrangian trajectory).
For the advection equation that means the outside value is used if n · v < 0 and the
inside value is used if n · v > 0. 252 CHAPTER 14. FINITE ELEMENT METHODS Figure 14.19: Flow ﬁeld and initial condition for Gaussian Hill experiment Proceeding as before and deﬁning the local/elemental approximation to Th as
N T (x) = Ti (t)φi (x) (14.168) i=1 and the degrees of freedom Ti are determined by the following ODE:
M dT
dt + G=0 Mji =
Gj E =
E (14.169) φi φj dV (14.170) φi G · n dS (14.171) where M is the local mass matrix; no assembly is required. The great value of DGM
is that the variational statement operates one element at a time, and hence the mass
matrix arising from the timederivative is purely local. The only linkage to neighboring
elements comes from considering the ﬂuxes along the boundary. Thus, even if the matrix
M is full, it is usually a small system that can be readily inverted, and the process of time
integration becomes considerably cheaper. Another beneﬁt of DGM is that it satisﬁes the
local conservation property so dear to many oceanographic/coastal modeler, since the
ﬂuxes deﬁned on each edge are unique. Finally, by dropping the continuity requirements
it becomes quite feasible to build locally and dynamically adaptive solution strategies
without worrying about global continuity of the function. In the next section we compare
the performance of the DGM with that of the continuous formulation on several problems;
all simulations will be performed using the spectral element interpolation. 14.6.1 Gaussian Hill Experiment The ﬁrst experiment is designed to establish the convergence property of the diﬀerent
methods. To this end we choose the classical problem of advecting a passive tracer in 14.6. THE DISCONTINUOUS GALERKIN METHOD (DGM) 253 the unit square domain (0 ≤ x, y ≤ 1) by a rotating nondivergent ﬂow given by:
u = −ω (y − 1/2), v = ω (x − 1/2). (14.172) where ω is set to 2π . The initial condition is inﬁnitely smooth and given by a Gaussian
distribution
r2
12
12
φ = e− l2 , r =
+ y−
(14.173)
x−
4
2
with an efolding length scale l = 1/16. Periodic boundary conditions are imposed on
all sides. All models were integrated for one rotation which time the solution should be
identical to the initial condition.
The convergence curves for the l2 norm of the error are displayed in 14.20 for the
diﬀerent formulations. The time integration consists of RK4 for traditional Galerkin
method ( which we will refer to as the CGM), and DGM; the time step was chosen so
that spatial errors are dominant. The convergence curves for CGM, and DGM are similar
and indicate the exponential decrease of the error as the spectral truncation is increased
for a constant elemental partition.
In order to compare the beneﬁts of h versus p reﬁnements, we plot in the right panels
of ﬁgure 14.20 the l2 error versus the total number of collocation points in each direction.
The beneﬁts of preﬁnement for this inﬁnitely smooth problem is readily apparent for
CG: Given a ﬁxed number of collocation points, the error is smallest for the smallest
number of elements, and hence highest spectral truncation. (The number of collocation
points is given by K (N − 1)+1, where N is the number of points per element and K is the
number of elements). The situation is not as clear for DGM where the diﬀerent curves
tend to overlap. This is probably due to the discontinuous character of the interpolation:
adjacent elements are holding duplicate information about the solution, and the total
number of degrees of freedom grows like KN . 14.6.2 Cone Experiment The second experiment is designed to establish the behavior of the diﬀerent methods in
the presence of “mild” discontinuities. The ﬂow ﬁeld is the same as for the Gaussian hill,
but the initial proﬁle consists of a cone:
u(x, y, t = 0) = max(0, 1 − 8r ), r = x− 1
4 2 + y− 1
2 2 . (14.174) The cone has a peak at r = 0 which decreases linearly to 0; there is a slope discontinuity
at r = 1/8. The initial conditions contours are similar to the one depicted in ﬁgure 14.19.
The same numerical parameters were used as for the Gaussian Hill problem.
The presence of the discontinuity ruins the spectral convergence property of the
spectral element method. This is born out in the convergence curves (not shown) which
display a 1/N convergence rate only in the l2 norm for a ﬁxed number of elements;
h reﬁnement is a more eﬀective way to reduce the errors in the present case. In the
following, we compare the performance of the diﬀerent schemes using a single resolution,
10x10 elemental partition with 6 points per element. Figure 14.21 compares the solution 254 CHAPTER 14. FINITE ELEMENT METHODS ε vs number of points per element ε vs total number of degrees of freedom 2 2 −2 −2 10 10
2 −4 −4 3 10 10 2 4
5 −6 −6 10 2 3 ε ε 2 10 8 −8 −8 10 10 4 10
−10 5 −10 10 10 8
6 8 10 12
N 14 16 18 0 ε vs number of points per element 50 100
Ndf 150 10
200 ε vs total number of degrees of freedom 2 2 −2 −2 10 10
2 −4 −4 10 10
3 2 4
−6 −6 2 10 2 10 ε ε 5 3 −8 −8 10 10 10 8 4 −10 −10 10 10 10
6 8 10 12
N 5
14 16 18 0 50 100
Ndf 8
150 Figure 14.20: Convergence curve in the L2 norm for the Gaussian Hill initial condition using, from top
to bottom, CGM, and DG. The labels indicate the number of elements in each direction. The abcissa on
the left graphs represent the spectral truncation, and on the right the total number of collocation points. 200 14.6. THE DISCONTINUOUS GALERKIN METHOD (DGM) 255 Figure 14.21: Contour lines of the rotating cone problem after one revolution for CG (left), and DG
(right), using the 10×6 grid. The contours are irregularly spaced to highlight the Gibbs oscillations. for the 4 schemes at the end of the simulations. We note that the contour levels are
irregularly spaced and were chosen to highlight the presence of Gibbs oscillations around
the 0level contour.
For CG, the oscillation are present in the entire computational region, and have
peaks that reaches −0.03. Although the DG solution exhibits Gibbs oscillations also,
these oscillations are conﬁned to the immediate neighborhood of the cone. Their largest
amplitude is one third that observed with CG. Further reduction of these oscillation
require the use of some form of dissipation, e.g. Laplacian, high order ﬁlters, or slope
limiters. We observe that CG shows a similar decay in the peak amplitude of the cone
with DG doing a slightly better job at preserving the peak amplitude. Figure 14.22
shows the evolution of the largest negative T as a function of the grid resolution for CG
and DG. We notice that the DG simulation produces smaller negative values, up to a
factor of 5, than CG at the same resolution. 256 CHAPTER 14. FINITE ELEMENT METHODS 0 10 −1 10
−min(T) 5
7 5 9
7
9 −2 10 −3 10 0 10 1 10
K 2 10 Figure 14.22: Min(T) as a function of the number of element at the end of the simulation. The red
lines are for DG and the blue lines for CG. The number of points per element is ﬁxed at 5, 7 and 9 as
indicated by the lables. Chapter 15 Linear Analysis
15.1 Linear Vector Spaces In the following we will use the convention that bold roman letter, such as x, denote
vectors, greek symbols denote scalars (real or complex) and capital roman letters denote
operators. 15.1.1 Deﬁnition of Abstract Vector Space We call a set V of vectors a linear vector space V if the following requirements are
satisﬁed:
1. We can deﬁne an addition operation, denoted by ‘+’, such that for any 2 elements
of the vector space x and y, the result of the operation z = (x + y) belongs to V .
We say that the set V is closed under addition. Furthermore the addition must
have the following properties:
(a) commutative x + y = y + x
(b) associative (x + y) + z = x + (y + z)
(c) neutral element: there exist a null zero vector 0 such that x + 0 = x
(d) for each vector x ∈ V there exist a vector y such that x + y = 0, we denote
this vector by y = −x.
2. We can deﬁne a scalar multiplication operation deﬁned between any vector x ∈ V
and a scalar α such that αx ∈ V , i.e. V is closed under scalar multiplication. The
following properties must also hold for 2 scalars α and β , and any 2 vectors x and
y:
(a) α(β x) = (αβ )x
(b) Distributive scalar addition: (α + β )x = αx + β x
(c) Distributive vector addition: α(x + y) = αx + αy
(d) 1x = x
(e) 0x = 0
257 258 CHAPTER 15. LINEAR ANALYSIS 15.1.2 Deﬁnition of a Norm In order to provide the abstract vector space with the sense of length and distance, we
deﬁne the norm or length of a vector x, as the number x . In order for this number to
make sense as a distance we put the following restrictions on the deﬁnition of the norm:
1. αx = α x
2. Positivity x > 0, ∀x = 0, and x = 0 ⇔ x = 0
3. Triangle or Minkowski inequality x + y ≤ x + y
With the help of the norm we can deﬁne now the distance between 2 vectors x and
y as x − y , i.e. the norm of their diﬀerence. So 2 vectors are equal or identical if
their distance is zero. Furthermore, we can now talk about the convergence of a vector
sequence. Speciﬁcally, we say that a vector sequence xn converges to x as n → ∞ if for
any ǫ > 0 there is an N such that xn − x < ǫ ∀n > N . 15.1.3 Deﬁnition of an inner product It is generally usefull to introduce the notion of angle between 2 vectors x and y. We
thus deﬁne the scalar (x, y) which must satisfy the following properties
1. conjugate symmetry: (x, y) = (y, x), where the overbar denotes the complex conjugate.
2. linearity: (αx + β y, z) = α(x, z) + β (y, z)
3. positiveness: (x, x) > 0, ∀x = 0, and (x, x) = 0 if x = 0.
The above properties imply the Schwartz inequality:
(x, y) ≤ (x, x) (y, y) (15.1) This inequality suggest that the inner product (x, x) can be deﬁned as a norm, so that
x = (x, x)1/2 With the deﬁnition of a norm we call 2 vectors orthogonal iﬀ (x, y) = 0.
Moreover iﬀ (x, y) = x y the 2 vectors are colinear or aligned. 15.1.4 Basis A set of vectors e1 , e2 , . . . , en diﬀerent from 0 are called a basis for the vector space V if
they have the following 2 properties:
1. Linear independence:
N
i=1 αi ei = 0 ⇔ αi = 0∀i (15.2) If at least one of the αi is nonzero, the set is called linearly dependent, and one of
the vectors can be written as a linear combination of the others. 15.1. LINEAR VECTOR SPACES 259 2. Completeness Any vector z ∈ V can be written as a linear combination of the
basis vectors.
The number of vectors needed to form a basis is called the dimension of the space V .
Put in another way, V is N dimensional if it contains a set of N independent vectors
but no set of (N + 1) of independent vectors. If N vectors can be found for each N , no
matter how large, we say that the vector space is inﬁnite dimensional.
A basis is very usefull since it allows us to describe any element x of the vector space.
Thus any vector x can be “expanded” in the basis e1 , e2 , . . . , en as:
N x= αi ei
i = α1 e1 + α2 e2 + . . . + αn en (15.3) This representaion is also unique by the independence of the basis vectors. The question
of course if how to ﬁnd out the coeﬃcients αi which are nothing but the coordinates of
x in ei . We can take the inner product of both sides of the equations to come up with
the following linear system of algebraic equations for the αi :
α1 (e1 , e1 ) +α2 (e2 , e1 ) + . . . +αn (en , e1 )
α1 (e1 , e2 ) +α2 (e2 , e2 ) + . . . +αn (en , e2 )
.
.
. = (x, e1 )
= (x, e2 ) (15.4) α1 (e1 , en ) +α2 (e2 , en ) + . . . +αn (en , en ) = (x, en )
The coupling between the equations makes it hard to compute the coordinates of a
vector in a general basis, particularly for large N . Suppose however that the basis set
is mutually orthogonal, that is every every vector ei is orthogonal to every other vector
ej , that is (ei , ej ) = 0 for all i = j . Another way of denoting this mutual orthogonality
is by using the Kronecker delta function, δij :
δij = 1 i=j
0 i=j (15.5) The orthogonality property can then be written as (ei , ej ) = δij (ej , ej ), and the basis is called orthogonal. For an orthogonal basis the system reduces to the uncoupled
(diagonal) system
(ej , ej )αj = (x, ej )
(15.6)
and the coordinates can be computed easily as
αj = (x, ej )
(x, ej )
=
(ej , ej )
ej 2 (15.7) The basis set can be made orthonormal by normalizing the basis vectors (i.e. rescaling
ej by such that their norm is 1), then αj = (x, ej ). 260 CHAPTER 15. LINEAR ANALYSIS 15.1.5 Example of a vector space Consider a set V whose elements are deﬁned by N tuples, i.e. each element x of V
is identiﬁed by N scalar ﬁelds (ξ1 , ξ2 , . . . , ξn ) Let us deﬁne the addition of 2 vectors x,
deﬁned as above, and y deﬁned by the N tuples (η1 , η2 , . . . , ηn ) as the vector z whose
N tuples are σi = ξi + ηi . Furthermore, we deﬁne the vector z = αa by its N tuples
σi = αξi . It is then easy to verify that this space endowed with the vector addition and
scalar multiplication deﬁned above fullﬁll the requirements of a vector space.
A norm for this vector space can be easily deﬁned by the socalled p norm where
1
p N x p p =
i=1 ξ  (15.8) A particularly usefull norm is the 2norm (p = 2), also, called the Euclidean norm. Other
usefull norms are the 1norm (p = 1) and the inﬁnity norm:
x ∞ = lim x
p→∞ p = max ξj .
1≤j ≤N (15.9) An inner product for this vector space can be deﬁned as:
N ξi η i (x, y) = (15.10) i=1 It can be easily veriﬁed that this inner product satiﬁes all the needed requirements.
Furthermore, the norm introduced by this inner product is nothing but the 2norm
mentioned above.
An orthonormal basis for the vector space V is given by;
e1 = (1, 0, 0, 0, . . . , 0)
e2 = (0, 1, 0, 0, . . . , 0)
.
.
. (15.11) en = (0, 0, 0, 0, . . . , 1)
is a complete and independent vector set. Thus V is N dimensional. 15.1.6 Function Space Spaces where the vectors are functions occupy an important place in linear analysis.
Their properties as linear spaces are harder to manipulate as they are inﬁnite dimensional
spaces. For example the space of all continuous functions deﬁned on the interval a ≤ t ≤ b
and which we denote by C (a, b) is a linear vector space where vector addition and scalar
multiplication are deﬁned in an obvious way. The inner product on this vector space is
the continuous analogue of the inner product deﬁned for the N tuple space. Suppose for
a moment that the functions x(t) and y(t) are deﬁned on N equally spaced points on 15.1. LINEAR VECTOR SPACES 261 the interval [a, b] (i.e. discrete space) by their pointwise values xi and yi at the points
ti , and let us deﬁne the discrete inner product as the Riemann type sum:
b−a
N (x, y) = N (15.12) xi yi
i=1 In the limit N tends to inﬁnity the above discrete sum become
N b−a
N →∞ N b (x, y) = lim xi yi =
i=1 x(t)y(t) dt. (15.13) a Two functions are said to be orthogonal if (x, y) = 0.
Similarly we can deﬁne the pnorm of a function as:
b x p =
a x(t)p dt (15.14) The 1norm, 2norm and ∞ norms follow by setting p = 1, 2. and ∞.
The main diﬃculties with function spaces are twofold. First they are inﬁnite dimensional and thus require an inﬁnite set of vectors to deﬁne a basis; proving completeness
is hence diﬃcult. Two, functions are usually deﬁned in a continuum where limits may
be in or outside the vector space. Some interesting issue arise. Consider for example the
sequence of functions 0, −1 ≤ t ≤ 0 1
(15.15)
x(t) =
nt,
0≤t≤ n 1 1,
≤t≤1
n deﬁned on C (−1, 1). This sequence converges to the step function, also known as the
Heaviside function, H (t) which does not belong to C (−1, 1) since it is discontinuous.
That although the sequence xn is in C (−1, 1), its limit as n → ∞ is not; we say that
the space does not contain its limit points and hence is not closed. This is akin to the
space C (−1, 1) having holes in it. This is rather unfortunate as closed spaces are more
tractable.
It is possible to create a closed space by changing the deﬁnition of the space slightly
to that of the Lebesgue space L2 (a, b), namely the space of functions that are square
integrable on the interval [a, b], i.e.
b x 2 =
a x(t)2 dt < ∞. (15.16) L2 (a, b) is an example of a Hilbert space – a closed inner product space with the
1
1
x = (x, x) 2 – a closed inner product space with the x = (x, x) 2 .
The issue of deﬁning a basis function for a function space is complicated by inﬁnite
dimension of the space. Assume I have an inﬁnite set of linearly independent vectors, if
I remove a single element of that set, the set is still inﬁnite but clearly cannot generate
the space. It turns out that it is possible to prove completeness but we will defer the
discussion until later. For the moment, we assume that it is possible to deﬁne such a 262 CHAPTER 15. LINEAR ANALYSIS basis. Furthermore, this basis can be made to be orthogonal. The Legendre polynomials
Pn are an orthogonal set of functions over the interval [−1, 1], and the trigonometric
functions einx are orthogonal over [−π, π ].
Suppose that an orthogonal and complete basis ei (t) has been deﬁned, then we can
expand a vector in this basis function:
x= ∞ αi ei (15.17) i=1 The above expansion is referred to as a generalized Fourier series, and the αi are the
Fourier coeﬃcients of x in the basis ei . We can also follow the procedure outlined for the
ﬁnitedimensional spaces to compute the αi ’s by taking the inner product of both sides
of the expansion. The determination of the coordinate is, again, particularly easy if the
basis is orthogonal
(x, ei )
(15.18)
αi =
(ei , ei )
In particular, if the basis is orthonormal αi = (x, ei ).
In the following we show that the Fourier coeﬃcients are the best approximation to
the function in the 2norm, i.e. the coeﬃcients αi minimize the norm x − i αi ei 2 .
We have:
x− αi ei 2
2 i = (x − i = (x, x) − αi ei , x − αi ei ) αi (x, ei ) − i (15.19) i αi αj (ei , ej(15.20)
) αi (ei , x) +
i i j The orthogonality of the basis functions can be used to simply the last term on the right
hand side to i αi 2 . If we furthermore deﬁne ai = (x, ei ) we have:
2
2 = x 2 x = x 2
2 (15.21) [(ai − αi )(ai − αi ) − ai ] (15.22) ai − αi 2 − (15.23) + i = [αi αi − ai αi − ai αi ] i αi ei i
i x− +
+ i a2
i Note that since the ﬁrst and last terms are ﬁxed, and the middle term is always greater
or equal to zero, the left hand side can be minimized by the choice αi = ai = (x, ei ). The
minimum norm has the value x2 − i ai 2 . Since this value must be always positive
then
ai  ≤ x2 ,
(15.24)
i A result known as the Bessel inequality. If the basis set is complete and the minimum
norm tend to zero as the number of basis functions increases to inﬁnity, we have:
i ai  = x2 , (15.25) which is known as the Paserval equality; this is a generalized “Pythagorean Theorem”. 15.2. LINEAR OPERATORS 15.1.7 263 Pointwise versus Global Convergence Consider the 2 functions x(t) and y(t) deﬁned on [−1, 1] as follows:
x(t) = t
y(t) = (15.26)
t,
1, for t > 0
x=0 (15.27) Both functions belong to C (−1, 1) and are identical for all t except t = 0. If we use the
2norm to judge the distance between the 2 functions, we get that x − y 2 = 0, hence
the functions are the same. However, in the maximum norm, the 2 functions are not
identical since x − y ∞ = 1. This example makes apparent that the choice of norms
is critical in deciding whether 2 functions are the same or diﬀerent, 2 functions that
may be considered identical in one norm can become diﬀerent in an another norm. This
is simply an apparent contradiction and reﬂects the fact that diﬀerent norms measure
diﬀerent things. The 2norm for example looks at the global picture and asks if the 2
functions are the same over the interval; this is the socalled meansquare convergence.
The inﬁnity norm on the other hand measures pointwise convergence. 15.2 Linear Operators An operator or (transformation) L, we mean a mapping from one vector space, called
the domain D , to another vector space called the range R. L thus describes functions
operating on vectors. As with functions, we require L to be uniquely valued although
not necessarily onetoone, i.e. we may have Lx = Ly = z for x = y. However,
Lx = Ly = z implies x = y, we say that L is onetoone, i.e. each vector z in R has a
single corresponding x such that Lx = z. Two operators A and B are equal if they have
the same domain and if Ax = B y, ∀x, y. Finally, we say that I is the identity operator
if I x = x, ∀x and Ø is the null operator if Øx = 0∀x.
Operation addition and multiplication by a scalar can be deﬁned. The operator
C = A + B is deﬁned as C x = Ax + B y, and C = αA is deﬁned as C x = α(Ax). The
product of two operators C = AB is deﬁned as C x = A(B x). We can easily see that
operator addition commutes (since vector addition must commute), whereas AB need
not equal BA, when they do, we say that A and B commute.
We call an operator linear if
L(αx + β y) = αLx + βLy (15.28) An operator is bounded if there is a positive constant c such that Lx < c, ∀x ∈ D .
The smallest suitable bound is called the norm of the operator, thus:
L = lubx=0 Lx
x (15.29) An adjoint to the operator L is the operator L∗ such that
(Lx, y) = (x, L∗ y), ∀x, y (15.30) 264 CHAPTER 15. LINEAR ANALYSIS It is easy to show the following properties:
(L∗ )∗ = L
∗ (15.31)
∗ (15.32) ∗ (A + B ) ∗
∗ (15.33) = A +B ∗ (AB ) = B +A If L∗ = L we call the operator self adjoint. 15.3 Eigenvalues and Eigenvectors Non trivial solutions x = 0 to the equation
Lx = λx, (15.34) are called eigenvectors, the scalars λ are called eigenvalues. The statement above asks
essentially if there are special vectors which when transformed by L produce parallel
vectors, the ratio of lengths of these two vectors is the eigenvalue. The above equation
can be restated as ﬁnd the nontrivial solution to the homogeneous equation
(L − λI )x = 0. (15.35) Usually the eigenvalues and eigenvectors occur in pairs. If the operator L is a matrix
the λ’s can be determined by solving the characteristic equations det(L − λI ) = 0.
The following results are very important:
1. The eigenvalues of a self adjoint operator are all real, and the eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.
2. For selfadjoint operator L on a ﬁnite dimensional domain V , k mutually orthogonal
eigenvectors can be found for each eigenvectors of multiplicity k.
3. The 2 properties above imply that the eigenvectors of a selfadjoint operators form
a basis for the ﬁnitedimensional space V . The situation is substantially more
complicated for inﬁnite dimensional spaces, and is taken care of by the Sturm
Liouville theory. 15.4 SturmLiouville Theory We will change our notation and drop the bold face of the vector notation. Given the 2
functions f and g on the interval a ≤ t ≤ b, we deﬁne ﬁrst an inner product of the form:
b f (t)g(t)w(t) dt (f, g) = (15.36) a where w(t) > 0 on a < t < b (this is a more general inner product that the one deﬁned
earlier which correspond to w(t) = 1). Let the operator L be deﬁned as follows:
d
dy
1
p(t)
+ r (t)
w(t) dt
dt
αy (a) + βy ′ (a) = 0, γy (b) + δy ′ (b) = 0
Ly = (15.37)
(15.38) 15.4. STURMLIOUVILLE THEORY 265 1.4 2 1.2
1.5 1
1 0.8
0.5 0.6 0 0.4 −0.5 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 15.1: Left: step function ﬁt f = 1; blue line is for N = 1, black line for N = 7,
and red line for N = 15. Right: Fourier expansion to f = esin πx , blue line is for N = 1,
red for N = 2, and black for N = 3; the circles show the function f In the present case L is a diﬀerential operator subject to homogeneous boundary conditions. It is easy to show that the operator L is self adjoint under the inner product
deﬁned in 15.36. Furthermore, the eigenvalue problem deﬁned by
Ly + λy = 0 (15.39) is called a SturmLiouville system. SturmLiouville systems have the crucial theorem
that
Theorem 2 If both p(t) and w(t) are analytic and positive over the interval [a, b], where
a and b are ﬁnite, then the eigenfunctions of the SturmLiouville system are complete
over L2 (a, b).
Completeness is also known to hold under other circumstances. For example if p(t)
vanishes at one or both end points, then the boundary conditions can be replaced with
the requirement that y or y ′ be ﬁnite there. If it happens that p(a) = p(b) we can replace
the boundary conditions by periodicity conditions y (a) = y (b) and y ′ (a) = y ′ (b).
Note carefully that the domain of the Sturm Liouville system is the set of function
that are twice diﬀerentiable and that satisfy certain boundary conditions, and yet the
theorem asserts completeness over a much broader space L2 , which contains functions
that need not be even continuous and that need not satify the boundary conditions.
The various special forms of the Sturm Liouville problems gives rise to various commonly known Fourier series. For example the choice of w = p = 1 and r = 0 gives rise
to the Fourier trigonometric series. For w = 1, p = 1 − t2 we get the Fourier Legendre
series, etc...
Example 18 The function y (t) = 1 on 0 ≤ t ≤ 1 can be expanded in the trigonometrics
series N =1 fm sin(mπx). It is easy to verify that
m
1 (sin mπx, sin kπx) =
0 1
sin kπx sin mπxdx = δkm
2 (15.40) 266 CHAPTER 15. LINEAR ANALYSIS
m
0
1
2
3
4
5
6
7
8
9
10
11 Am
0.000000000000000
1.13031820798497
4.968164449421373×10−18
4.433684984866381×10−02
3.861917048379140×10−17
5.429263119140263×10−04
3.898111864745672×10−16
3.198436462370136×10−06
3.363168791013840×10−16
1.103677177269183×10−08
4.748619297285671×10−17
2.497959777896152×10−11 Bm
0.532131755504017
4.097072104153782×10−17
0.271495339534077
5.790930955238388×10−17
5.474240442093724×10−03
2.457952426063564×10−17
4.497732295430222×10−05
5.047870580187706×10−17
1.992124806619515×10−07
1.091272736411650×10−16
5.505895970193617×10−10
1.281761515677824×10−16 Table 15.1: Fourier coeﬃcients of example 19
4
and that the Fourier coeﬃcients are given by: fm = 0, for even m, and fm = mπ , for
odd m. Figure 15.1 illustrates the convergence of the series as the number of Fourier
functions retained, N , increases. Example 19 The function f = esin πx is periodic over the interval [−1, 1] and can be
expanded into a Fourier series of the following form:
N f (x) = Am cos mπx + Bm sin mπx (15.41) m=0 The Fourier coeﬃcients can be determined from:
Am = 1
sin πx cos mπxdx
−1 e
,
1
2
−1 cos mπxdx Bm = 1
sin πx sin mπxdx
−1 e
1
2
−1 sin mπxdx (15.42) Since the integrals cannot be evaluated analytically, we compute them numerically using
a very high order methods. The ﬁrst few Fourier coeﬃcients are listed in table 15.1.
Notice in particular the rapid decrease of Am  and Bm  as m increases, and the fact
that with 3 Fourier modes the series expansion and the original function are visually
identical. 15.5 Application to PDE The method of separation variables relies on the Sturm Liouville theory to generate a
basis made up of eigenfunctions for the function space where the solution is sought.
The problem boils down to ﬁnding the Fourier coeﬃcients of the solution in that basis.
The type of eigenfunctions used depends on the geometry of the domain, the boundary
conditions and the partial diﬀerential equations. 15.5. APPLICATION TO PDE 267 Example 20 Let us take the example of the Laplace equation ∇u = 0 deﬁned on the
rectangular domain 0 ≤ x ≤ a and 0 ≤ y ≤ b, and subject to the boundary conditions
that u = 0 on all boundaries except the top boundary where u(x, y = b) = v (x). Separation of variables assumes that the solution can be written as the product of functions
that depend on a single independent variable: u(x, y ) = X (x)Y (y ). When this trial
solution is substituted in the PDE, we can derive the identity:
Yyy
Xxx
=−
X
Y (15.43) Since the left hand side is a function of xonly, and the right hand side a function of y
only, and the equality must hold for arbitrary x and y the two ratios must be equal to a
constant which we set to −λ2 , and we end up with the 2 equations
Xxx + λ2 X = 0 (15.44) 2 (15.45) Yyy − λ Y =0 Notice that the 2 equations are in the form of a SturmLiouville problem. The solutions
of the above 2 systems are the following set of functions:
X = A cos λx + B sin λx (15.46) Y (15.47) = C cosh λy + D sinh λy where A, B , C and D are integration constants. Applying the boundary conditions at
x = 0 and y = 0, we deduce that A = C = 0. The boundary condition at x = b produces
the equation
B sin λa = 0
(15.48)
The solution B = 0 is rejected since this would result in the trivial solution u = 0, and
hence we require that sin λa = 0 which yields the permissible values of λ:
λn = nπ
, n = 1, 2, . . . .
a (15.49) There are thus an inﬁnite number of possible λ which we have tagged by the subscript
n, with correponding Xn (x), Yn (x), and unknown constants. Since the problem is linear
the sum of these solution is also a solution and hence we set
u(x, y ) = ∞ En sin λn x sinh λn y (15.50) n=1 where we have set En = Bn ∗ Dn . The last unused boundary condition determines the
constants En with the equation
v (x) = ∞ En sin λn x sinh λn b (15.51) n=1 It is easy to show that the functions sin λn x are orthogonal over the interval 0 ≤ x ≤ a,
i.e.
b
a
(15.52)
sin λn x sin λm x dx = δnm
2
a 268 CHAPTER 15. LINEAR ANALYSIS which leads to
En = 2
a b
a sin λn x v (x) dx. (15.53) The procedure outlined above can be reintrepeted as follows: the sin λn x are the
eigenfunctions of the Sturm Liouville problem 15.44 and the eigenvalues are given by
15.49; the basis function is complete and hence can be used to generate the solution as
in equation 15.50; the coordinates of the solution in that basis are determined by 15.53.
The eigenvalues and eigenfunctions depend on the partial diﬀerential equations, the
boundary conditions, and the shape of the domain. The next few examples illustrate
this dependence.
Example 21 The heat equation ut = ν (uxx + uyy ) in the same domain as example 20
subject to homogeneous Dirichlet boundary conditions on all sides will generate the eigenπx
values eigenfunction pairs sin ma , in the xdirection and sin nπy in the y direction.
b
The solution can be written as the double series:
u(x, y, t) = ∞ ∞ Amn sin n=1 m=1 nπy −αmn t
mπx
sin
e
, αmn = −
a
b m2 π 2 n 2 π 2
+2
a2
b (15.54) Changing the boundary conditions to homogeneous Neumann conditions in the y direction
will change the eigenfunctions to cos( nπy ). If the boundary conditions is homogeneous
b
Neumann at the bottom and Dirichlet at the top, the eigenpairs in the y directions
become cos (2n+1)πy .
b
Example 22 Solution of the wave equation utt = c2 ∇2 u in the disk 0 ≤ r ≤ a using
cylindrical coordinates will produce the following set of equations in each independent
variable:
Ttt + κ2 c2 T
2 Θθθ
22 =0 (15.55) +λ Θ = 0 (15.56) 2 2 r Rrr + rRr + (κ r − λ )R = 0 (15.57) Since the domain is periodic in the azimuthal direction, we should expect a periodic
solution and hence λ must be an integer. The radial equation is nothing but the Bessel
equation in the variable κr , its solutions are given by R = An Jn (κr ) + Bn Yn (κr ). Bn = 0
must be imposed if the solution is to be ﬁnite at r = 0. The eigenvalues κ are determined
by imposing a boundary condition at r = a. For a homogeneous Dirichlet conditions,
the eigenvalues are determined by the roots ξmn of the Bessel functions Jn (ξm ) = 0, and
hence κmn = ξmn /a. Note that κmn is the radial wavenumber. The solution can now be
expanded as:
u(r, θ, t) = ∞ ∞ Jn (κmn r ) [(Amn cos nθ + Bmn sin nθ ) cos σmn t + n=0 m=0 (Cmn cos nθ + Dmn sin nθ ) sin σmn t] (15.58) 15.5. APPLICATION TO PDE 269 where σmn = κmn c is the time frequency. The integration constants must be determined
from the initial conditions of which there must be 2 since we have a second derivative
in time. In the present examples the radial eigenfunctions are given the Bessel functions
of the ﬁrst kind and order n and the eigenvalues are determined by the roots of the Jn .
Notice that the Bessel equation is also a Sturm Liouville problem and hence the basis
Jn (κmn r ) must be complete and orthogonal. Periodicity in the azimuthal direction yields
the trigonometric functions, and quantizes the eigenvalues to the set of integers. 270 CHAPTER 15. LINEAR ANALYSIS Chapter 16 Rudiments of Linear Algebra
16.1 Vector Norms and Angles We need to generalize the notion of distance and angles for multidimensional systems.
These notions are intuitive for two and three dimensional vectors and provide a basis for
the generalization. This generalization does not prescribe a single formula for a norm
or distance but lists the properties that must be satisﬁed for a measure to be called a
distance or angle. 16.1.1 Vector Norms A vector norm is deﬁned as the measure of a vector in realnumber space, i.e. it is a
function that associated a real (positive) number for each member of the vector space.
The norm of a vector u is denoted by u and must satisfy the following properties:
1. positivity: u ≥ 0, and if u = 0, then u = 0. All norms are positive and only
the null vector has 0 norm.
2. Scalar multiplication: αu = α u , for any scalar α.
3. triangle inequality: u + v ≤ u + v for any two vectors u and v.
For a vector u = (u1 , u2 , . . . , uN ) an operator that deﬁnes a norm is the socalled Lp
norm deﬁned as:
1
p N u p =
i=1 The following are frequently used values for p:
1norm: p = 1, u 1 = N ui ;
i=1
2norm: p = 2, u 2 =
maxnorm: p = ∞, u ∞ N
2
i=1 ui  ; = maxi ui .
271 ui p (16.1) 272 CHAPTER 16. RUDIMENTS OF LINEAR ALGEBRA 16.1.2 Inner product An inner product is an operation that associates a real number between any two vectors
u and v. It is usually denoted by (u, v) or u · v; and has the following properties:
1. commutative: (u, v) = (v, u).
2. linear under scalar multiplication: (αu, v) = α(u, v).
3. linear under vector addition: (u, v + w) = (u, v) + (u, v).
4. positivity (u, u) ≥ 0, and (u, u) = 0 implies u = 0.
The norm and inner product deﬁnition allow us to deﬁned the cosine of an angle between
two vectors as:
(u, v)
cos θ =
(16.2)
uv
The properties of the inner product allows it to deﬁned a vector norm, often called the
innerproduct induced norm: u = (u, u). 16.2 Matrix Norms Properties of a matrix norm:
• positivity: L > 0∀L, L = 0 implies L = 0.
• scalar multiplication: αL = α L , where α is a scalar
• triangle inequality: L + M ≤ L + M
• LM ≤ L M
Since matrices and vectors occur together, it is reasonable to put some conditions on
their respective norms. Matrix and vector norms are compatible if:
Lu ≤ L u ∀ u =0 (16.3) It is possible to use a vector norm to deﬁne a matrix norm, the socalled, subordinate
norm:
Lu
= max Lu
(16.4)
L = max
u
u
u =1
Here are some common matrix norms some satisfy the compatibility and can be regarded
as subordinate norms:
1. 1norm L
sum.
2. ∞norm L
sum. 1 = maxj (
∞ N
i=1 lij ) = maxi ( This is also referred to as the maximum column
N
j =1 lij ) This is also referred to as the maximum row 3. 2norm L 2 = ρ(LT L) where ρ is the spectral radius (see below). If the matrix
L is symmetric LT = L, then L 2 = ρ(L2 ) = ρ(L). 16.3. EIGENVALUES AND EIGENVECTORS 16.3 273 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors of a matrix L are the pairs (λ,u) such that
Lu = λu, u = 0 (16.5) The eigenvalues can be determined by rewriting the deﬁnition as (L − λi I )ui = 0, and
requiring that the solutions to this equation have nontrivial solution. The condition is
that the determinant of the matrix be zero: det(L − λI ) = 0. This is the characteristic
equation of the system. For an N × N matrix, the characteristic equation is an ndegree
polynomial that admits n complex roots. These roots are precisely the eigenvalues of the
system.
Assuming, the eigenvalues of the matrix L are λi and ui , then the following properties
hold:
• The transpose of the matrix, LT has the same eigenvalues and eigenvectors as the
matrix L.
• The matrix L2 = LL has the same eigenvectors as L and has the eigenvalues λ2 .
i
• If the inverse of the matrix exist, L−1 , then its eigenvectors are the same as those
of L and its eigenvalues are 1/λi .
• If B = a0 + a1 L + a2 L2 + . . . + ap Lp , is a polynomial in L, then it has the same
eigenvectors as L and its eigenvalues are a0 + a1 λ + a2 λ2 + . . . + ap λp .
• If L is a real symmetric matrix, (LT = L), then eigenvectors corresponding to
distinct eigenvalues are mutually orthogonal, and all its eigenvalues are real. 16.4 Spectral Radius The spectral radius, ρ(L), of a matrix L is given by its largest eigenvalue in magnitude:
ρ(L) = max(λi ) (16.6) i The spectral radius is the lowest for all compatible matrix norms. The proof is simple.
Let the matrix L have the eigenvalues and eigenvectors λi , ui . Then
≤L Lui ui (16.7) λi ui = λi  ui
and hence λi  ≤ L . This result holds for all eigenvalues, and in particular for the
eigenvalue with the largest magnitude, i.e. the spectral radius:
ρ(L) = max(λi ) ≤ L
i (16.8) The above result holds for all matrix norms. For the case of the 1 or ∞norms, we
have Gershgorin ﬁrst theorem, that the spectral radius is less then the largest sum of the
absolute values of the row or columns entries, namely: ρ(L) ≤ L 1 and ρ(L) ≤ L ∞ . 274 CHAPTER 16. RUDIMENTS OF LINEAR ALGEBRA Gershgorin’s second theorem puts a limit on where to ﬁnd the eigenvalues of a matrix
in the complex plane. Each eigenvalue is within a circle centered at li i, the diagonal entry
of the matrix, and of radius R:
N λi − li,i  ≤ 16.5 j =1,j =i lij  = li,1  + l2,i  + . . . li,i−1  + li,i+1  + . . . li,N  (16.9) Eigenvalues of Tridiagonal Matrices A tridiagonal matrix is a matrix whose only nonzero entries are on the main diagonal,
and on the ﬁrst upper and lower diagonals. l1,1 l1,2 l 2,1 L= 0
.
.
. 0 0 0 l2,2 l2,3
l3,2 l3,3
··· 0 0
.
.
. 0
l3,4
..
. 0 lN,N −1 lN,N (16.10) Tridiagonal matrices occur often in the discretization of onedimensional partial diﬀerential equations. The eigenvalues of some of these tridiagonal matrices can sometimes
be derived for constant coeﬃcients PDE. In this case the matrix takes the form: ab
c a b ca L= b
.. b . c (16.11) a
ca The eigenvalues and corresponding eigenvectors are given by √
λi = a + 2 bc cos for i = 1, 2, . . . , N . iπ
,
N +1 ui = c
b
c
b
c
b
c
b 1
2 2
2 j
2 N
2 iπ
sin
N +1
2iπ
sin
N +1
.
.
.
sin
.
.
.
sin jiπ N +1 N iπ N +1 (16.12) 16.5. EIGENVALUES OF TRIDIAGONAL MATRICES 275 For periodic partial diﬀerential equations, the tridiagonal system is often of the form ab
c a b ca
L= b c
b
..
c . b (16.13) a
ca The eigenvalues are then given by:
λi = a + (b + c) cos 2π (i − 1)
2π (i − 1)
+ i(c − b) sin
N
N (16.14) A ﬁnal useful result is the following. If a real tridiagonal matrix has either all its oﬀdiagonal element positive or all its oﬀdiagonal element negative, then all its eigenvalues
are real. 276 CHAPTER 16. RUDIMENTS OF LINEAR ALGEBRA Chapter 17 Programming Tips
17.1 Introduction The implementation of numerical algorithm requires familiarity with a number of software packages and utilities. Here is a short list of the minimum required to get started
in programming:
1. Basics of the Operating System such as ﬁle manipulation. The RSMAS library has
UNIX: the basics. The library has also a bunch of electronic books on the subject.
Two titles I came across are: Unix Unleashed, and Learning the Unix Operating
System: Nutshell Handbook
2. Text editor to write the computer program. Most Unix books would have a short
tutorial on using either vi or emacs for editing text ﬁles. There are a number of
simpler visual editors too such as jed. Web sites for vi or its close cousin vim are:
• http://www.asu.edu/it/fyi/dst/helpdocs/editing/vi/
This is actually a very concise and fast introduction to vi. Start with it and
then go to the other web sites for more indepth information.
• http://docs.freebsd.org/44doc/usd/12.vi/paper.html
• http://www.vim.org/ To learn about emacs, and its visual counterpart, xemacs, visit
• http://www.math.utah.edu/lab/unix/emacs.html
This seems like a good and brief introduction so that you can be editing ﬁles
with simple commands.
• http://cmgm.stanford.edu/classes/unix.emacs.html • http://www.lib.chicago.edu/keith/tclcourse/emacstutorial.html • http://www.xemacs.org/
This is the Grapher User Interface version of emacs. It is much like notepad
in WINDOWS in that the commands can be entered visually.
277 278 CHAPTER 17. PROGRAMMING TIPS 3. Knowledge of a programming language. Fortran is still the preferred language for
numerical programming, although C and C++ have been used in certain applications requiring sophisticated data structures. Section 17.2 has an example to
introduce various elements of the language.
4. Compiler to turn the program into machine instructions. Section 17.3 discusses
compiler issues and how to use the compiler options in helping you to track errors
in the coding.
5. Debugger to track down bugs in a computer program.
6. Visualization software to plot the results of computations.
7. Various mathematical libraries such as LAPACK for linear algebra routines and
FFTPACK for Fast Fourier Transform. 17.2 Fortran Example The following is a simple fortran 90 code that makes use of simple language syntax such
as declaring variable, looping, opening a ﬁle, and writing data out. You should type
in the program to practice the editor commands, compile it and run it. Then use a
visualization package to plot the results.
!
! This is a sample fortran program to introduce the language.
! It divides the interval [1 1] into M points and computes the
! functions sin(x) and cos(x). The results are written to the
! terminal and to a file called waves.dat
!
! Comments are marked with an exclamation mark; the compiler
! ignores all characters after the "!" sign.
!
! A fortran statement that does not fit into a single line can
! be continued on the next one by terminating it with a "&" sign.
program waves
! name of program unit
implicit none ! prevents compiler from assigning default types
! and forces user to declare every variable !.Variable Declaration starts here
integer, parameter :: M=21 ! declares the value of a constant
! that does not change throughout
! the calculation
integer :: i
! declares an integer counter
real, parameter :: xmin=1.0, xmax=1.0 ! single precision constants
real :: f(M)
! real array with M entries 17.2. FORTRAN EXAMPLE 279 real :: pi,x,y,dx
!.End of Variable Declaration
! Executable statements are below ! Unit 6 is the terminal also called stdout
write(6,*)’Hello World’
! Open a file to write out the data.
open(unit=9,
& ! file unit is number 9
file=’waves.out’, & ! output file name is waves.out
form=’formatted’, & ! data written in ASCII
status=’unknown’, & ! create file if it does not exit already
action=’write’)
! file meant for writing
pi = 2.0*asin(1.0)
! initialize pi
dx = (xmaxxmin)/real(M1)
! gridsize
do i = 1,M
! counter: starts at 1, increments by 1 and ends at M
!...indent statements within loop for clarity
x = (i1)*dx + xmin ! location on interval
y = sin(pi*x)
! compute function 1
f(i) = cos(pi*x)
! compute function 2
write(6,*) x,y
! write two columns to terminal
write(9,*) x,y,f(i) ! write three columns to file
enddo
! end of do loop must be marked.
close(9) ! close file (optional) write(6,*)’Done’
stop
end program waves
!
!
! Compiling the program and creating an executable called waves:
!
$ f90 waves.f90 o waves
! If "o waves" is ommited the executable will be called a.out
! by default. The fortan 90 compiler (f90) may have a different name
! on your system. Possible alternatives are pgf90 (Portland Group
! compiler), ifc (Intel Fortran Compiler), and xlf90 (on IBMs).
!
!
! Running the program
!
$ waves 280 CHAPTER 17. PROGRAMMING TIPS !
!
! Expected Terminal output is:
! Hello World
! 1.000000
8.7422777E08
! 0.9000000
0.3090170
! 0.8000000
0.5877852
! 0.7000000
0.8090170
! 0.6000000
0.9510565
! 0.5000000
1.000000
! 0.4000000
0.9510565
! 0.3000000
0.8090171
! 0.2000000
0.5877852
! 9.9999964E02 0.3090169
! 0.0000000E+00 0.0000000E+00
! 0.1000000
0.3090171
! 0.2000000
0.5877854
! 0.3000001
0.8090171
! 0.4000000
0.9510565
! 0.5000000
1.000000
! 0.6000000
0.9510565
! 0.7000000
0.8090169
! 0.8000001
0.5877850
! 0.9000000
0.3090170
!
1.000000
8.7422777E08
! Done
!
!
! Visualizing results with matlab:
!
$ matlab
!> z = load(’waves.out’);
% read file into array z
!> size(z)
% get dimensions of z
!> plot(z(:,1),z(:,2),’k’) % plot second column versus first in black
!> hold on;
% add additional lines
!> plot(z(:,1),z(:,3),’r’) % plot third column versus first in red
!> xlabel(’x’);
% add labels to xaxis.
!> ylabel(’f(x)’);
% add labels to yaxis.
!> title(’sine and cosine curves’);
% add title to plot
!> legend(’sin’,’cos’,0)
% add legend
!> print depsc waves
% save results to a color
!>
% encapsulated postscript file
!>
% called waves.eps. The extension
!>
% eps will be added automatically.
! Viewing the postscript file:
!
$ ghostscript waves.eps 17.3. DEBUGGING AND VALIDATION 281 !
! Printing the file to color printer
!
$ lpr Pmpocol waves.eps 17.3 Debugging and Validation Errors are invariably introduced when implementing a numerical algorithm in a computer
code. There are actually two kinds of errors to watch for:
1. Algorithmic Errors: are conceptual in nature and are independent of the actual
computer code. These kinds of errors can usually be studied theoretically when the
algorithm is devised. Examples include stability and convergence of a numerical
scheme to its mathematically continuous form. It is usually very hard to solve
algorithmic problems with computer experimentation. The latter can only be a
guide and/or a conﬁrmation of theory.
2. Programming Errors: are introduced when translating an algorithm into actual
computer code. These errors are often referred to as bugs and there are techniques
that makes tracking them simpler. This is what we will concentrate on in this
section. 17.3.1 Programming Rules The following are rules of thumb devised to help a programmer write “good” code. The
primary advice is to write clear readable code that is easy to validate and maintain.
Other advice towards that goal are:
• Write modular programs with clearly deﬁned functional unit. By separating the
diﬀerent steps of an algorithm, it becomes easier to “abstract” the code, and to
make connections with the theory.
• Each modular unit should be validated. By gaining conﬁdence in the basic working
of the building units it becomes easier to cobble together more complicated programs. Moreover, some basic units can be reused in new programs without having
to rewrite them.
• Comment the program to explain the diﬀerent steps. Choose meaningfull names
for variables. Document the input and output of subroutines and their basic tasks.
• Write clear code, and do not worry about eﬃciency in either CPU or memory.
Modern computers are vastly superior to the ones common during the early years
of computing. Memory and CPU speed constraints forced programmers to use
coding tricks that obfuscated the code.
• Improve the readability of the code. Do not be afraid to leave white spaces. Indent
do loops, logical statements, and functional units so they are easy to identify.
• Last but not least make sure you are using the appropriate algorithm for what you
intend to do. 282 CHAPTER 17. PROGRAMMING TIPS 17.3.2 Coding tips and compiler options If you write programs in the “right” way, you will have few syntax errors, a good compiler
(called with the right options) will ﬂag them for you, and you will correct them easily. It
is important to emphasize that most compilers are rather lenient by default, and must
be invoked with special options in order to get many useful warnings.
Some of these compiler options enable compiletime checks, i.e. the compiler will spot
errors during the translation of the code to machine language and prior to executing the
program. Examples include: disabling implicit type declarations, and ﬂagging standard
language violations. Others options enable various runtime checks (when the code is
actually executed) such as: checking array bounds, trapping ﬂoatingpoint exceptions,
and special variable initializations that are supposed to produce ﬂoatingpoint exceptions
if uninitialized variables are used.
The following is a list of programming tips that will help minimize the number of
bugs in the code.
• Do not used implicitly declared types. In fortran variables whose name start with
the letters i,j,k,l,m,n are integers by default and all others are reals by default.
Use the statement implicit none in your code to force the compiler to list every
undeclared variable, and to catch mistyped variables. It is also possible to do that
using compiler options (see the manual pages for the speciﬁc options, on UNIX
machines it is usually u).
• Make sure every variable is initialized properly before using it, and do not assume
that the compiler does it automatically. Some compiler will allow you to initialize
variables to a bogus value, a Nan (short for not a number, so that the code trips if
that variable is used in an operation before overwriting the Nan.
• Use the include header.h statement to make sure that common blocks are identical across program units. The common declaration would thus go into a separate
ﬁle (header.h) which can be subsequently included in other subroutines without
retyping it.
• Check that the argument list of a call statement matches in type and number the
argument list of the subroutine declaration.
• Remarks on the pitfalls list The improved syntax of Fortran 90 eliminates
some common programming pitfalls. Note that a Fortran 90 compiler will often
only be able to help you if you make use of the stricter checking features of the
new standard:
1. IMPLICIT NONE
2. Explicit interfaces
3. INTENT attributes
4. PRIVATE attributes for modulewide data that should not be accessible to
the outside 17.3. DEBUGGING AND VALIDATION 283 • Use list ﬁles to catch compiler report. When compiling the compiler throws rapidly
a list of errors at you, being out of context they are hard to understand, and even
harder to remember when you return to the editor. Using two windows one for
editing and one for debugging is helpful. You can also ask the compiler to generate
a LIST FILE, that you can look at (or print) in a separate window.
• Use modules and interface to double check the argument lists of calling subroutines. 17.3.3 Run time errors and compiler options Some bugs cannot be caught at compile time and produce errors only when the code is
executed. The compiler switches can help you catch some common bugs. Here is again
a list of tips.
1. Do not optimize the code in the ﬁrst run. Rather compile it with a debugging
option (usually g) to produce trace back information. The program can then let
you know the statement number that caused the fatal error.
2. Do array bound checking. The code will crash if you are trying to access memory
beyond that available for an array. The common ﬂag for this is C but changes
from compiler to compiler. This ﬂag will slow down the performance of the code.
However, you ﬁrst concern should be a correct code rather then fast code.
3. Some ﬂoating point operations can result in a Nan. The code should stop then and
issue an error report. Various switches trap diﬀerent ﬂoating point exceptions.
You want to catch division by zero, overﬂows (where the number is too large
to be represent in the machine precision), underﬂows (similar but for very small
numbers). Underﬂows are not as problematic as overﬂows and can usually be
ignored. Check the manual for the right compiler switches.
4. Test routines individually to check if they are working according to their speciﬁcation. Try ﬁrst “trivial” cases where you know the answers to check the results you
get.
5. Use print statements liberally to spot check the value of variables.
6. Use a symbolic debugger to trace the execution of your program.
The ﬁrst rule about debugging is staying cool, treat the misbehaving program as an
intellectual exercise, or as a detective work.
1. You got some strange results that made you think there is a bug. Think again, are
you sure they are not the correct output for some special input?
2. If you are not sure what causes the bug, DON’T try semirandom code modiﬁcations, that’s seldom works. Your aim should be to gather as much as possible
information! 284 CHAPTER 17. PROGRAMMING TIPS 3. If you have a modular program, each part does a clearly deﬁned task, so properly placed ’debug statements’ can ISOLATE the malfunctioning procedure/codesection.
4. If you are familiar with a debugger use it, but be careful not to be carried away by
the many options and start playing. 17.3.4 Some common pitfalls • Datatypes
1. Using implicit variable declarations
2. Using a nonintrinsic function without a type declaration
3. Incompatible argument lists in call and the routine deﬁnition
4. Incompatible declarations of a common block
5. Using constants and parameters of incorrect (smaller) type
6. Assuming that untyped integer constants get typed properly
7. Assuming intrinsic conversionfunctions take care of result type
• Arithmetic
1. Using constants and parameters of incorrect type
2. Uncareful use of automatic type promotions
3. Assuming that dividing two integers will give a ﬂoatingpoint result.
4. Assuming integer exponents (e.g. 2**(3)) are computed as ﬂoatingpoint
numbers
5. Using ﬂoatingpoint comparisons tests, .EQ. and .NE. are particularly risky
6. Loops with a REAL or DOUBLE PRECISION control variable
7. Assuming that the MOD function with REAL arguments is exact
8. Assuming realtointeger assignment will work in all cases
• Miscellaneous
1. Code lines longer than the allowed maximum
2. Common blocks losing their values while the program runs
3. Aliasing of dummy arguments and common block variables, or other dummy
arguments in the same subprogram invocation
4. Passing constants to a subprogram that modiﬁes them
5. Bad DOloop parameters (see the DO loops chapter)
6. TABs in input ﬁles  what you see is not what you get!
• General 17.3. DEBUGGING AND VALIDATION 285 1. Assuming variables are initialized to zero
2. Assuming variables keep their value between the execution of a RETURN
statement and subsequent invocations
3. Letting array indexes go out of bounds
4. Depending on assumptions about the computing order of subexpressions
5. Assuming shortcircuit evaluation of expressions
6. Using trigonometric functions with large arguments on some machines
7. Inconsistent use of physical units 286 CHAPTER 17. PROGRAMMING TIPS Chapter 18 Debuggers
A debugger is a very useful tool for developing codes, and in ﬁnding and ﬁxing bugs
introduced in the code development. Most compiler providers include a debugger as
part of their software bundle. Since we are using the Portland Group compiler for the
class, the discussion here will center on its debugger primarily, even though a lot of the
information applies equally to other compiler/debugger systems. Check the manual for
the speciﬁc compiler/debugger in case of problems. 18.1 Preparing the code for debugging A debuggable executable must have extra information embedded in it for debugging
purposes. The debugger then makes use of this information to report the values of
variables, and allow the programmer to follow the execution of the program line by line.
Compiler options must be used to instruct the compiler to embed this information in the
object ﬁles. Here is a useful subset of these options pertinent to the PG compiler:
1. g Generate information for the debugger, this is necessary on most computers to
enable the printing of useful debugging information. Avoid using any optimization
in the code development phase (so avoid the O options). Occasionaly you want to
debug with optimization on, the gopt would then be necessary.
2. C or Mbounds: generate code to check array bounds
3. Ktrap=listofoptions: helps trap ﬂoating point problems. The list is a comma
separates list of strings that controls which ﬂoating point operations to catch. These
include:
(a) Ktrap=divz: trap divide by zero.
(b) Ktrap=ovef: trap ﬂoating point overﬂows.
(c) Ktrap=unf: trap underﬂow (number too small to be representable).
(d) Ktrap=inv: trap invalid operands (e.g. square root of negative numbers).
A commonly useful subset is to set Ktrap=divz,inv,ovf
287 288 CHAPTER 18. DEBUGGERS 4. Mchkptr: Check if code mistakenly references NULL pointers.
5. Mchkstk: Check the stack for available space upon entry to and before the start
of a parallel region. Useful when many private variables are declared.
Compiler
Enable Debugging
Bounds Checking
Uninitialized variables
Floating point trap
Floating point stack 18.2 PGI
g
C
NA
Ktrap
Mchkstk Intel
g
CB
ftrapuv
fpe0
fpstkchk gcc
g
C
ﬀpetrap IBM
g
C
qinitauto
qﬂttrp Pathscale
g Running the debugger The command pgdbg executable wil run the code executable under the control of the
debugger. The default is to start the debugger under a Graphical User Interface (GUI).
If you have a slow connection, the GUI can slow down your work because of the graphics
overhead. Most debugger include a command, text line interface. For PG it is invoked
with the text option”. The command will start the debugger which in turn will “load”
the executable and all its pertinent information. The debugger then passes control to
the user and waits for further instructions. Most debuggers come with an online help
facility to allow users to learn the command list interactively, the command to initiate
is usually called help. Here we list brieﬂy a subset of these commands and that have
proved to be most useful for the author.
The debugger GUI is fairly intuitive; it will open up at least one window to display
source lines, and another one for I/O operations. The GUI and text versions of the
debugger can be controlled via command lines. Here we will cover some the most useful
one, and we refer the user to the manual for further information.
1. run Will cause the code to start executing.
2. list 10,40 will list the lines 10,40, list without argument will list lines 10 to
20 from the current statement. 3. stop at xyz will put a breakpoint at line xyz. Execution will stop at the line so
the user can examine variables. 4. print sn will print the value of the variable sn. 5. assign sn=expression assigns the expression expression to the variable sn. 6. whatis sn will report the data type of variable sn. 7. where will report the current statement. 8. step will execute the next source line, including stepping into a function or subroutine. 18.2. RUNNING THE DEBUGGER
9. 289 stepi will execute a single intruction (as opposed to the entire source line. step
¡count¿ will execute ¡count¿ instructions. 10. display list the expressions being printed at breakpoints. display <exp1>,<exp2>
prints <exp1> and <exp2> at every breakpoint. 11. next will cause the debugge to skip over a function or subroutine, i.e. executing
it in its entirety. 12. continue will cause the execution to resume from the point it stopped to the next
breakpoint. 290 CHAPTER 18. DEBUGGERS Bibliography
Arakawa, A., 1966. Computational design for long term numerical integration of the
equations of ﬂuid motion: twodimensional incompressible ﬂow. part i. Journal of
Computational Physics 1 (1), 119–143.
Arakawa, A., Hsu, Y.J., 1981. Energy conserving and potentialenstrophy dissipating
schemes for the shallowwater equations. Monthly Weather Review 118, 1960–1969.
Arakawa, A., Lamb, V. R., 1977. Computational Design of the Basic Dynamical Processes
of the UCLS general circulation model. Vol. 17. Academic Press, New York, p. 174.
Arakawa, A., Lamb, V. R., 1981. A potential enstrophy and energy conserving scheme
for the shallow water equations. Monthly Weather Review 109, 18–36.
Balsara, D. S., Shu, C.W., 2000. Monotonicity preserving weighed essentially nonoscillatory schemes with increasingly high order of accuracy. Journal of Computational
Physics 160, 405–452.
Boris, J. P., Book, D. L., 1973. Flux corrected transport, i: Shasta, a ﬂuid transport
algorithm that works. Journal of Computational Physics 11, 38–69.
Boris, J. P., Book, D. L., 1975. Flux corrected transport, ii: Generalization of the method.
Journal of Computational Physics 18, 248.
Boris, J. P., Book, D. L., 1976. Flux corrected transport, iii: Minimum error fct algorithms. Journal of Computational Physics 20, 397.
Boyd, J. P., 1989. Chebyshev and Fourier Spectral Methods. Lecture Notes in Engineering. SpringerVerlag, New York.
Butcher, J. C., 1987. The Numerical Analysis of Ordinary Diﬀerential Equations. John
Wiley and Sons Inc., NY.
Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., 1988. Spectral Methods in
Fluid Dynamics. Springer Series in Computational Physics. SpringerVerlag, New York.
Dormand, J. R., 1996. Numerical Methods for Diﬀerential Equations, A Computational
Approach. CRC Press, NY.
Dukowicz, J. K., 1995. Mesh eﬀects for rossby waves. Journal of Computational Physics
119, 188–194.
291 292 BIBLIOGRAPHY Durran, D. R., 1999. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, New York.
Finlayson, B. A., 1972. The Method of Weighed Residuals and Variational Principles.
Academic Press.
Jiang, C.S., Shu, C.W., 1996. Eﬃcient implementation of weighed eno schemes. Journal
of Computational Physics 126, 202–228.
Karniadakis, G. E., Sherwin, S. J., 1999. Spectral/hp Element Methods for CFD. Oxford
University Press.
Leonard, B. P., MacVean, M. K., Lock, A. P., 1995. The ﬂux integral method fo multidimensional convection and diﬀusion. Applied Mathematical Modelling 19, 333–342.
Shchepetkin, A. F., McWilliams, J. C., 1998. Quasimonotone advection schemes based
on explicit locally adaptive dissipation. Montlhy Weather Review 126, 1541–1580.
Shu, C.W., 1998. Essentially nonoscillatory and weighed essentially nonoscillatory
schemes for hyperbolic conservation laws. Springer, New York, p. 325.
Suresh, A., Huynh, H. T., 1997. Accurate monotonicity preserving schemes with rungekutta time stepping. Journal of Computational Physics 136, 83–99.
Whitham, G. B., 1974. Linear and Nonlinear Waves. WileyInterscience, New York.
Zalesak, S. T., 1979. Fully multidimensional ﬂuxcorrected transport algorithms for ﬂuids. Journal of Computational Physics 31, 335–362.
Zalesak, S. T., 2005. The design of ﬂuxcorrected transport algorithms for structured
grids. In: Kuzmin, D., L¨hner, R., Turek, S. (Eds.), FluxCorrected Transport.
o
Springer, pp. 29–78. ...
View
Full
Document
This note was uploaded on 01/08/2012 for the course MPO 662 taught by Professor Iskandarani,m during the Spring '08 term at University of Miami.
 Spring '08
 Iskandarani,M
 The Land

Click to edit the document details