High Order Explicit Two-Step Runge-Kutta Methods for Parallel Computers

Abstract

In this paper we study a class of explicit pseudo two-step Runge-Kutta methods (EPTRK methods) with additional weights v. These methods are especially designed for parallel computers. We study s-stage methods with local stage order s and local step order s + 2 and derive a sufficient condition for global convergence order s + 2 for fixed step sizes. Numerical experiments with 4- and 5-stage methods show the influence of this superconvergence condition. However, in general it is not possible to employ the new introduced weights to improve the stability of high order methods. We show, for any given s-stage method with extended weights which fulfills the simplifying conditions B(s) and C(s - 1), the existence of a reduced method with a simple weight vector which has the same linear stability behaviour and the same order.