Lax equivalence theorem

For a consistent finite-difference approximation to a well-posed linear initial-value problem, stability is equivalent to convergence as the discretization is refined.

Consistency (numerical analysis): A numerical scheme is consistent when its discrete equations approximate the differential equation as the discretization is refined. The theorem assumes consistency before stability can guarantee convergence.
Lax–Richtmyer stability: Stability for a numerical approximation of an evolution equation, expressed through uniform bounds on its solution operators. This operator-bound formulation is central to the theorem's stability-to-convergence argument.
Peter D. Lax: An American mathematician known for major contributions to partial differential equations and numerical analysis. Lax co-formulated the equivalence theorem with Robert D. Richtmyer.
Von Neumann stability analysis: A Fourier-mode method for testing stability of linear finite-difference schemes with constant coefficients. It often checks the stability hypothesis needed to invoke the theorem.
Nonlinear stability: The study of how perturbations behave in numerical approximations of nonlinear equations. The classical theorem does not directly cover nonlinear initial-value problems.
Stability (numerical analysis): A numerical scheme is stable when perturbations in its data or intermediate computations remain controlled under refinement. Under the theorem's assumptions, stability supplies the missing condition for convergence.
Local truncation error: The residual produced when an exact solution is substituted into a discrete numerical scheme. Consistency makes this one-step defect vanish as the mesh is refined.
Robert D. Richtmyer: An American mathematician and physicist whose work helped establish modern numerical methods for partial differential equations. Richtmyer co-authored the theorem's influential 1956 formulation.
Upwind scheme: A finite-difference method for transport equations that chooses spatial differences according to the direction of wave propagation. Its stability and consistency conditions illustrate the theorem in a concrete scheme.
Finite element method: A numerical method that approximates solutions by combining simple functions over a mesh of elements. Its convergence theories use frameworks beyond the theorem's original finite-difference statement.

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