Unformatted text preview: c(n+1); ; cn;) + t K(2cn+ + c(n+1); ) = t (2ln + l(n+1);)
M( c + 2
; cn+ + t K(cn+ + 2c(n+1);) = t (ln + 2l(n+1);):
This pair of equations must be solved simultaneously for the two unknown solution vectors
cn+ and c(n+1); . This is an implicit Runge-Kutta method. Problems 1. Consider the Galerkin method in time with a continuous basis as represented by
(9.3.1). Assume that the solution c(t) is approximated by the linear function
(9.3.2a-c) on (tn tn+1) as in Example 9.3.1, but do not assume that the test space
w(t) is linear in time.
1.1. Specifying w( ) = !( ) 1 1 : : : 1]T
and assuming that M and K are independent ot t, show that (9.3.1a) is the
weighted average scheme M + tK]cn+1 = M ; (1 ; ) tK]cn + t (1 ; )ln + ln+1]
with R !( )N ( )d
1 1 1
+1 ;1 When di erent trial and test spaces are used, the Galerkin method is called a
1.2. The entire e ect of the test function !(t) is isolated in the weighting factor .
Furthermore, no integration by parts was performed, so that !(t) need not be
continuous. Show that the choices of !(t) listed in Table 9.3.1 correspond to
the cited methods.
2. The discontinuous Galerkin method may be derived by simultaneously discretizing
a partial di erential system in space and time on
(t ; n; t(n+1); ). This form
may have advantages when solving problems with rapid dynamics since the mesh
may be either moved or regenerated without concern for maintaining continuity 18 Parabolic Problems Scheme
Forward Euler (9.2.10b)
(1 + ) 0
( ) 1/2
Backward Euler (9.2.11b) (1 ; ) 1
Nn+1( ) 2/3
Table 9.3.1: Test functions ! and corresponding methods for the nite element solution
of (9.2.4a) with a linear trial function.
between time steps. Using (9.2.2a) as a model spatial nite element formulation,
assume that test functions v(x y t) are continuous but that trial functions u(x y t)
have jump discontinuities at tn. Assume Dirichlet boundary data and show that
the space-time discontinuous Galerkin form of the problem is
(v ut)ST + (v( tn) u( tn+) ; u( tn;)) + AST (v u) = (v f )ST
8v 2 H01( (tn+ t(n+1); ))
and (v u)ST = Zt ; (n+1) ZZ tn+ vudxdydt AST (v u) = (vx pux)ST + (vy puy )ST + (v qu)ST : In this form, the nite element problem is solved on the three-dimensional strips
(tn; t(n+1); ), n = 0 1 : : : . 9.4 Convergence and Stability
In this section, we will study some theoretical properties of the discrete methods that
were introduced in Sections 9.2 and 9.3. Every nite di erence or nite element scheme
for time integration should have three properties:
1. Consistency: the discrete system should be a good approximation of the di erential
2. Convergence: the solution of the discrete system should be a good approximation
of the solution of the di erential equation.
3. Stability: the solution of the discrete system should not be sensitive to small perturbations in the data. 9.4. Convergence and Stability 19 Somewh...
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