O teorema de equivalência de Lax-Richtmyer e sua aplicação

Abstract

The Lax-Richtmyer equivalence theorem establishes a crucial tool in the context of numerical analysis, simplifying the theoretical validation of finite difference schemes (FDS) — a family of algorithms for the numerical approximation of solutions of partial differential equations (PDE). A consistent FDS is deemed valid when we can prove both its stability and its convergence. The equivalence theorem states that, for a consistent FDS, convergence occurs if — and only if — we also have stability. This result is essential in the analysis of such schemes due to to the existence of numerous techniques for proving stability and the complexity of proving convergence. This work will analyze the mathematical basis of finite difference schemes, determining the consistency, stability, and convergence properties. We will also study Fourier analysis in the context of PDEs, present the equivalence theorem and its proof, and lastly, construct a brief example of its application.