Parallel-iterated pseudo two-step Runge-Kutta-Nyström methods for nonstiff second-order IVPs

The aim of this paper is to consider parallel iteration schemes for a general class of pseudo two-step Runge-Kutta-Nyström (RKN) methods of arbitrary high order for solving nonstiff initial-value problems y″(t) = f(y(t)), y(t0) = y0, y′(t0) = y0 on parallel computers. Starting with an s-stage pseudo two-step RKN method of order p* with w implicit stages, we apply the highly parallel PC iteration process in P(EC)mE mode. The resulting PIPTRKN method (parallel-iterated pseudo two-step RKN method) uses an optimal number of processors equal to w ≤ p*/2. By a number of numerical experiments, we show the superiority of the PIPTRKN methods proposed in this paper over both sequential and parallel methods available in the literature.