Improved Parallel-iterated Pseudo Two-step RKN Methods for Nonstiff Problems

The aim of this paper is to consider parallel PC iteration schemes for a general class of pseudo two-step RKN methods (PTRKN methods) of arbitrary high order and arbitrary low number of implicit stages for solving nonstiff second-order IVPs y"(t) = f(y(t)), y(t(0)) = y(0), y'(t(0)) = y'(0). Starting with an s-stage pseudo two-step RKN method of order p* with w implicit stages and applying the highly parallel PC iteration process lead to a parallel-iterated PTRKN method (PIPTRKN method) (in P(CE) mE mode). By replacing in PIPTRKN method, the old predictor with a new one, we are led to a PC method (in PE(CE) E-m mode) which will be called improved PIPTRKN method (IPIPTRKN method). The resulting IPIPTRKN method uses an optimal number of processors equal to w <= p*/2. By a number of numerical experiments, we show the superiority of the IPIPTRKN methods considered in this paper over both sequential and parallel methods available in the literature.