Some recent applications of asymptotic error expansions to finite-difference schemes

Abstract

In this paper we will present some results on asymptotic error expansions of the approximate solutions of differential equations by finite-difference schemes. We will present some quite well-known material, in an effort to make the presentation self-contained, as well as discuss recent work by Engquist on linear multistep methods for initial-value problems, by Kreiss on extrapolation procedures for elliptic finite-difference schemes, by Pereyra and co-workers on iterated deferred-correction methods for elliptic equations and by the author and Hald on the finite-element method. Practical aspects of the subject as well as the use of error expansions as a technical device in theoretical numerical analysis are discussed.