Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs

 The aim of this paper is to consider a parallel predictor–corrector (PC) iteration scheme for a general class of pseudo two-step Runge–Kutta methods (PTRK methods) of arbitrary high-order for solving first-order nonstiff initial-value problems (IVPs) on parallel computers. Starting with ans-stage pseudo two-step RK method of orderp ∗ withwimplicit stages, we apply a highly parallel PC iteration process inPE(CE) m Emode. The resulting parallel PC method can be viewed as a parallel-iterated pseudo two-step Runge–Kutta method (PIPTRK method) with an improved (new) predictor formula and therefore will be called the improved PIPTRK method (IPIPTRK method). The IPIPTRK method uses an optimal number of processors equal tow p ∗ /2. Numerical experiments show that the IPIPTRK methods proposed in this paper are superior to the efficient sequentialDOPRI5andDOP853 codes and parallel PIRK methods available in the literature.

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