Runge-Kutta-Nyström-type parallel block predictor-corrector methods

This paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nystrr om correctors. Our approach is to apply the predictor-corrector method not only with stepsize h, but, in addition (and simultaneously) with stepsizes a i h; i = 1; : : :; r. In this way, at each step, a whole block of approximations to the exact solution at oo-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange or Hermite interpolation. Since the block approximations at the oo-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nystrr om corrector methods, the computation of the approximation at each oo-step point is also highly parallel. By a number of widely-used test problems, a class of the resulting block predictor-corrector methods is compared with both sequential and parallel RKN methods available in the literature and is shown to demonstrate superior behaviour.