Explicit pseudo two-step Runge-Kutta methods for parallel computers

The aim of this paper is to investigate a class of explicit pseudo two-step Runge-Kutta methods of arbitrarily high order for nonstiff problems for systems of first-order differential equations. By using collocation techniques we can obtain for any given order of accuracy p, a stable pth-order explicit pseudo two-step Runge-Kutta method requiring only one effective sequential right-hand side evaluation per step on multiprocessor computers. 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.