A theory of explicit finite-difference schemes

Abstract: Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as systematic ways of matching up to the operator solution of the partial differential equation. By completely abandon the idea of approximating derivatives directly, the theory 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.