On the error structure of the implicit Euler scheme applied to stiff systems of differential equations
type: Item
instance of: scholarly article
title: On the error structure of the implicit Euler scheme applied to stiff systems of differential equations
zbMATH Open document ID: 0685.65062
DOI: 10.1007/BF02241856
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: This paper is concerned with the existence and behavior of asymptotic expansions of the global error when the implicit Euler scheme is applied to a linear, self-adjoint constant coefficient stiff system of differential equations with the stiff eigenvalues arbitrarily distributed on the negative real axis. It must be pointed out that in earlier papers on this subject the existence of asymptotic expansions in the form \(e(t)=\sum^{q}_{j=p}h^ je_ j(t)+O(h^{q+1})\) has already been proved but the size of the O-constant was not studied and therefore the existence of such expansions may be completely useless in practice. Here the existence of asymptotic expansions called asymptotically correct, i.e. with moderate O-constants, is addressed. The main result is that with the above assumptions an asymptotically correct expansion does not exist immediately after the initial point due to the existence of irregular error components, but these irregular terms are damped (algebraically) along the integration interval. To confirm the above results the author studies the behavior of the error for a linear problem with a matrix arising from the standard space discretization of the one- dimensional heat equation on a uniform mesh.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4123192
zbMATH Keywords: extrapolation methods
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: global error
zbMATH Keywords: implicit Euler scheme
zbMATH Keywords: stiff system
zbMATH Keywords: stiff eigenvalues
author: W. Auzinger
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02241856
OpenAlex ID: W1503195659
description: scientific article; zbMATH DE number 4123192