2007/09/26 by John Jossey, Jossey, John, Anil N. Hirani +1
Engineering · Mathematics · #65J05 #65J10 #65J15 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.0709.4046
openalex publication_date 2007/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration.