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IMA Journal of Numerical Analysis Advance Access published online on December 10, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn043
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Multistep cosine methods for second-order partial differential systems

B. Cano{dagger}

Departamento de Matemática Aplicada, Universidad de Valladolid, C/ Doctor Mergelina s.n., 47011 Valladolid, Spain

M. J. Moreta{ddagger}

Departamento de Fundamentos del Análisis Económico I, Facultad de Ciencias Económicas y Empresariales, Universidad Complutense de Madrid, Campus de Somosaguas, Pozuelo de Alarcón, 28223 Madrid, Spain

{dagger} Email: bego{at}mac.uva.es

{ddagger} Corresponding author. Email: mjesusmoreta{at}ccee.ucm.es

Received on 28 May 2007. Revised on 9 June 2008.


  Abstract

In this paper we construct and analyse some multistep methods that integrate exactly the stiff part of a second-order partial differential equation. Much emphasis is given to symmetric methods of this type in order to deal with Hamiltonian problems. In such a way, we obtain very efficient methods because they are explicit, stable and, in the case of symmetric methods, conserve properties of the system that they imitate, as is shown in a forthcoming paper. The Gautschi method is the simplest of these methods. We analyse it here using different techniques and assumptions from those in the literature, which also allow the study of methods of higher order. In particular, a symmetric fourth-order multistep method of this type is thoroughly constructed and analysed, including its resonances and possible ways to filter them.

Key Words: exponential integrators; multistep methods; second-order partial differential equations; stability; convergence; resonances


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