Premium
A unified derivation of finite‐difference schemes from solution matching
Author(s) -
Chin Siu A.
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21993
Subject(s) - mathematics , partial differential equation , matching (statistics) , finite difference , nonlinear system , finite difference method , first order partial differential equation , partial derivative , operator (biology) , mathematical analysis , repressor , quantum mechanics , gene , biochemistry , statistics , physics , chemistry , transcription factor
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