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function. However, consistency—the requirement that the finite difference method approximates the correct partial differential equation—is straightforward to verify, and stability is typically much easier to show than convergence (and would be needed in any event to show that
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The importance of the theorem is that while the convergence of the solution of the finite difference method to the solution of the partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a
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will not destroy the computation). Hence convergence is usually shown via the Lax equivalence theorem.
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Lax, P. D.; Richtmyer, R. D. (1956). "Survey of the
Stability of Linear Finite Difference Equations".
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