Oscillatory Störmer–Cowell methods

Abstract

We consider explicit methods for initial-value problems for special second-order ordinary differential equations where the right-hand side does not contain the derivative of y and where the solution components are known to be periodic with frequencies ω j lying in a given nonnegative interval [ ω ̄ , ω ̄ ] . The aim of the paper is to exploit this extra information and to modify a given integration method in such a way that the method parameters are “tuned” to the interval [ ω ̄ , ω ̄ ] . Such an approach has already been proposed by Gautschi in 1961 for linear multistep methods for first-order differential equations in which the dominant frequencies ω j are a priori known. In this paper, we only assume that the interval [ ω ̄ , ω ̄ ] is known. Two “tuning” techniques, respectively based on a least squares and a minimax approximation, are considered and applied to the classical explicit Störmer–Cowell methods and the recently developed parallel explicit Störmer–Cowell methods.