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Unformatted text preview: with a = 1 and 8
< ;2(1 + x) if ; 1 x < ;1=2
if ; 1=2 x < 1=2 :
(x) = : 2x
2(1 ; x) if 1=2 x < 1 Compute solutions on 0 < t 2 using J = 10 20 40 and = 1=2 1. (Values of
N follow from the Courant number.) Compare the accuracy of the solutions at
t = 2. Tabulate errors at t = 2 in the maximum and L2 norms and estimate
the order of convergence of the numerical to the exact solution. Plot the
solutions as a function of x at t = 2.
1.4. Repeat Part 3 using either the upwind or Lax-Friedrichs schemes. Compare
results with those obtained using the Lax-Wendro scheme. 3.4 The Lax Equivalence Theorem
In Section 3.3, we learned that every di erence scheme should be consistent, convergent,
and stable. The Lax Equivalence Theorem expresses a relationship between these three
properties. Theorem 3.4.1. (Lax Equivalence Theorem). Given a properly posed linear initial value problem and a consistent nite di erence approximation of it, then stability is necessary
and su cient for convergence.
Proof. We will prove that a consistent and stable di erence scheme is convergent. The
proof that unstable schemes do not converge is more involved and we'll defer it to Chapter
6. The entire proof appears in Richtmyer and Morton 3] and Strikwerda 5], Chapter
10. 28 Basic Theoretical Concepts Let Un be the solution of the di erence equation (3.1.7) and let
discretization error. Then
un+1 = L un + t n:
Let en n be the local un ; Un be the global discretization error and subtract (3.1.7) to obtain
en+1 = L en + t n: For simplicity, assume that L is independent of n and iterate the above relation to
en = (L )ne0 + t Ln;1 0 + Ln;2 1 + : : : + n;1]:
Taking a norm and noting that e0 = 0 in the absence of round-o errors, kenk t kLn;1 kk 0 k + kLn;2kk 1k + : : : + k ;1 k]: n (3.4.1) The di erence scheme is consistent, so we can make k arbitrarily small by making x
and t su ciently small. Let us choose a total time of interest T and an > 0 such
that k k k for all k t T whenever x, t < . Furthermore, the scheme is stable,
so there exists a constant C such that kLk k C for all k t T . With (3.4.1), these
considerations imply that kenk n tC : If n t T kenk = kun ; Unk T C and the scheme converges.
Remark 1. The Lax Equivalence Theorem provides the link between stability and
convergence that, perhaps, we expected based on the examples of Chapter 2.
Remark 2. In practice, convergence is di cult to establish, while consistency and
stability are less so. Consistency merely requires the use of Taylor's series expansions and
stability can be proven, at least, in simpli ed situations using von Neumann's approach.
Thus, Lax's theorem provides very useful information. Bibliography
1] D. Gottlieb and S.A. Orszag. Numerical Analysis of Spectral Methods: Theory and
Applications. Regional Conference Series in Applied Mathematics, No. 26. SIAM,
2] A.R. Mitchell and D.F. Gri ths. The Finite Di erence Method in Partial Di erential
Equations. John Wiley and Sons, Chichester, 1980.
3] R.D. Richtmyer and K.W. Morton. Di erence Methods for Initial Value Problems.
John Wiley and Sons, New York, second edition, 1967.
4] G. Strang. Linear Algebra and its Applications. Academic Press, New York, third
5] J.C. Strikwerda. Finite Di erence Schemes and Partial Di erential Equations. Chapman and Hall, Paci c Grove, 1989. 29...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14