A class of explicit parallel two-step Runge-Kutta methods

Abstract

The aim of the present paper is to construct a class of two-step Runge-Kutta methods of arbitrarily high order for application to parallel computers. Starting with an s-stage implicit two-step Runge-Kutta method of order p with k = p/2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta 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 here over parallel methods available in the literature.