Explicit symmetric Runge-Kutta-Nyström methods for parallel computers

In this paper, we are concerned with parallel predictor-corrector (PC) iteration of Runge-Kutta-Nyström (RKN) methods in P(EC)mE mode for integrating initial value problems for the special second-order equation y″(t) = f(y(t)). We consider symmetric Runge-Kutta-Nyström (SRKN) corrector methods based on direct collocation techniques which optimize the rate of convergence of the PC iteration process. The resulting PISRKN methods (parallel iterated SRKN methods) are shown to be much more efficient when they are compared to the PC iteration process applied to the Gauss-Legendre RKN correctors