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November 3, 1999, 13:49 
CFD Lax Equivalence Theorem?

#1 
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In the thoery of numerical ODEs the Lax Equivalence Theorem says that given a wellposed IVP(initialvalue problem) or IBVP (initial boundaryvalue problem) and a finitedifference scheme consistent with it stability and convergence are equivalent.
(1) Is there anything like this in CFD where the governing equations are generally nonlinear? (2) If not how can we justify using this theorem in CFD or generally speaking in PDEs? (3) Is there a relationship between this theorem and the fact that when our residuals go towards zero and the conclusion that our solution has converged? Thanks in advance to any 'takers'. 

November 3, 1999, 16:41 
Re: CFD Lax Equivalence Theorem?

#2 
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(1)There is no equivalent theorem for nonlinear equations. In fact, the theory for nonlinear pde's is extremely lacking. Existence cannot be proved for nonlinear equations like the Euler equations. If we assume existence, then the LaxWendroff theorem tells us about the uniqueness of the numerical scheme for scalar conservation laws. Stability is also a difficult thing to prove in the nonlinear case. People usually apply some linearisation and then prove the stability for the linearised equations. Such a procedure can only provide necessary conditions only. (2)Most of the numerical schemes and related concepts in cfd are first developed for the linear scalar case, and then extended in a straightforward way to nonlinear systems of equations. There is no reason why this should work, but it does almost all the time. In fact, it can be taken as an unwritten rule in cfd that a scheme which does not work in the linear scalar case will almost certainly fail in the nonlinear case. Hence the results concerning linear equations are necessary conditions for nonlinear equations, but may not be sufficient. (3)The decay of residuals in a numerical solution has nothing to do with uniqueness of the solution. Jameson has shown that the Euler equations can have different numerical solutions in the transonic case. Only for a scalar conservation law, the LaxWendroff theorem tells us that a numerical scheme that is consistent, conservative and stable will converge to a correct weak solution(for hyperblic equations).


November 4, 1999, 12:05 
Re: CFD Lax Equivalence Theorem?

#3 
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Thanks for your comments. I am familiar with the LaxWendroff scheme but not the LaxWendroff theorem. From your comments it looks like it might be related to the LaxEquivalence theorem. Can you tell me what it is?
In your third comment, you said that the decay of residuals in a numerical solution has nothing to do with the uniqueness of the solution. I do not agree with you entirely on this point. Let us assume we are solving the nonsingular matrix equation Ax = b. Let us define e = xx' where x is the exact solution and x' the approximate solution. Also, let the residual be defined as r = b  Ax' = A(xx')= Ae. It looks like as the residual goes to zero e goes to zero. As e goes to zero x'=e+x approaches the exact solution. Thus in this case as the residual goes to zero our numerical solution approaches the unique solution. It is for this and a few more examples why I do not agree with you on point (3). 

November 4, 1999, 13:18 
Re: CFD Lax Equivalence Theorem?

#4 
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(1)That is true. But we are talking of systems of nonlinear pde's. The corresponding numerical schemes are also nonlinear. In such a case, the Lax Equivalence theorem is a neccesary condition, but not sufficient to ensure convergence. The example you gave, Ax=b, is of course linear. (2)The LaxWendroff theorem states that a numerical scheme for the scalar conservation law, du/dt + df/dx = 0, converges to a unique (weak) solution, if it is consistent, stable and conservative. There is no equivalent theorem for systems of conservation laws.


November 4, 1999, 14:32 
Re: CFD Lax Equivalence Theorem?

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Thanks again. You are right that my question was specific to nonlinear pde's. However, your occasional references to linear methods coupled with your statement (3) were the reasons why I thought that your conclusion reflected in (3) included both linear and nonlinear pde's. Hence the reason for my linear example.
It does look like the LaxEquivalence theorem could be used to weed out unstable or inconsistent schemes. 

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