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-Nyström 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.