Lax equivalence theorem

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In numerical mathematics , Lax's equivalence theorem is the fundamental theorem in the analysis of the finite difference method for the numerical solution of partial differential equations . The claim is that for a correctly posed linear initial value problem, a consistent method is convergent if and only if it is stable .

The theorem means that the desired convergence of the solution of the finite difference method for the solution of the partial differential equation can only be determined with great difficulty, since the numerical solution is defined recursively . However, the consistency of the method, i.e. That is, the numerical method approximates the differential equation, easy to check, and stability is usually much easier to show than convergence (this would have to be proven anyway to show that rounding errors do not corrupt the solution). Therefore, convergence is usually shown using the equivalence theorem.

In this context, stability means that a matrix norm of the matrix that is used in the iteration is at most one . This is called (practical) Lax-Richtmyer stability. Instead, a Von Neumann stability analysis is often carried out, although a Von Neumann stability only implies a Lax-Richtmyer stability in certain cases.

The set is named after Peter Lax . Sometimes it is also referred to as the Lax-Richtmyer sentence after Peter Lax and Robert Richtmyer .

credentials

  1. ↑ John C. Strikwerda: Finite Difference Schemes and Partial Differential Equations . Chapman & Hall, 1989, pp. 26/222 (English).
  2. ^ GD Smith: Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. Oxford University Press, 1985, pp. 67-68 (English).
  3. ↑ PD Lax, RD Richtmyer: Survey of the stability of linear finite difference equations . Comm. Pure Appl. Math. 9 (1956), 267-293 MR0079204 doi : 10.1002 / cpa.3160090206 (English).

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