A unified derivation of finite‐difference schemes from solution matching

Abstract

Conventional finite‐difference schemes for solving partial differential equations are based on approximating derivatives by finite‐differences. In this work, an alternative method is proposed which views finite‐difference schemes as systematic ways of matching up to the operator solution of the partial differential equation. By completely abandoning the idea of approximating derivatives directly, the method provides a unified description of explicit finite‐difference schemes for solving a general linear partial differential equation with constant coefficients to any time‐marching order. As a result, the stability of the first‐order algorithm for an entire class of linear equations can be determined all at once. Because the method is based on solution‐matching, it can also be used to derive any order schemes for solving the general nonlinear advection equation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 243–265, 2016

The maximal order of semidiscrete schemes for quasilinear first order partial differential equations

“…The authors of this work conclude that the maximal order of linear schemes for which (1) is stable is 2r. In [2] the author derives conditions for linear, explicit time-marching methods approximating the m-th order linear equation with constant coefficients to any order.…”