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Linear stability analysis in the numerical solution of initial value problems

Published online by Cambridge University Press:  07 November 2008

J.L.M. van Dorsselaer ,
J.F.B.M. Kraaijevanger  and
M.N. Spijker
J.L.M. van Dorsselaer
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: spijker@rulcri.leidenuniv.nl
J.F.B.M. Kraaijevanger
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: spijker@rulcri.leidenuniv.nl
M.N. Spijker
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: spijker@rulcri.leidenuniv.nl
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Extract

This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.

Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.

The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.

The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.

Type
Research Article
Information
Acta Numerica , Volume 2 , January 1993 , pp. 199 - 237
Copyright
Copyright © Cambridge University Press 1993

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