The aim of this paper is to investigate a general class of explicit pseudo two-step Runge-Kutta-
Nystr??m 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 pth-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. ?? 1999 Elsevier Science Ltd. All rights reserved.
