Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Abstract

The paper describes the construction of explicit Runge-Kutta-Nyström (RKN) methods of arbitrarily high order. The order is borrowed from an underlying implicit RKN method. For the approximate solution of this method, an iteration scheme is defined. Prescribing a fixed number of iterations, the resulting scheme is an explicit RKN method. The iteration scheme is defined in such a way that many of the right-hand side evaluations can be done concurrently. As a result, explicit RKN schemes of order p are obtained which require, on a parallel computer, approximately p/2 right-hand side evaluations per step. Both in fixed- and variable-step mode, the schemes are compared with existing (sequentiasl) high-order RKN methods from the literature and are shown to demonstrate superior behaviour.