Explicit parallel two-step Runge-Kutta-Nyström methods

Abstract

The aim of this paper is to design a class of two-step Runge-Kutta-Nyström methods of arbitrarily high order for the special second-order equation y″( t ) = f(y( t )), for use on parallel computers. Starting with an s -stage implicit two-step Runge-Kutta-Nyström method of order p with k = p 2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P ( EC ) m E mode. In this way, we obtain an explicit two-step Runge-Kutta-Nyström method that has order p for all m and that requires k ( m + 1) right-hand side evaluations per step of which each k evaluation can be computed in parallel. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.