Oscillatory Störmer-Cowell methods

We consider explicit methods for initial-value problems for special second-order ordinary differential equations where the righthand side does not contain the derivative of ${bf y$ and where the solution components are known to be periodic with frequencies $|omega_j$ lying in a given nonnegative interval $[ under{omega, bar{omega ]$. 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 $[ under{omega, bar{omega ]$. 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 $omega_j$ are a priori known. In this paper, we only assume that the interval $[under{omega,bar{omega]$ is known. Two 'tuning' techniques, respectively based on a least squares and a minimax approximation, are considered and applied to the {it classical explicit St'ormer-Cowell methods and the recently developed {it parallel explicit St'ormer-Cowell methods.