Multistep cosine methods for second-order partial differential systems

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.