A general class of explicit pseudo two-step RKN methods on parallel computers

The aim of this paper is to investigate a general class of explicit pseudo two-step Runge-Kutta-Nystrom methods (RKN methods) of arbitrarily high order for nonstiff problems for systems of special second-order differential equations y "(t) = f(y(t)). Order and stability considerations show that we can obtain for any given p, a stable p(th)-order explicit pseudo two-step RKN method requiring p - 2 right-hand side evaluations per step of which each evaluation can be obtained in parallel. Consequently, on a multiprocessor computer, only one sequential right-hand side evaluation per step is required. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature. (C) 1999 Elsevier Science Ltd. All rights reserved.