Parallel diagonally implicit Runge-Kutta-Nyström methods

Abstract

In this paper, we study diagonally implicit iteration methods for solving implicit Runge-Kutta-Nyström (RKN) methods on parallel computers. These iteration methods are such that in each step, the iterated method can be regarded as a diagonally implicit Runge-Kutta-Nyström method (DIRKN method). The number of stages of this DIRKN method depends on the number of iterations and may vary from step to step. Since a large number of these stages can be computed in parallel, and since the total number of stages can be kept small by a suitable choice of the parameters in the iteration process, the resulting variable-stage DIRKN methods are efficient on parallel computers. By using implicit Runge-Kutta Nyström methods with high stage order, the phenomenon of order reduction exhibited in many problems with large Lipschitz constants does not deteriorate the accuracy of these variable-stage DIRKN methods. By a number of numerical experiments the superiority of the parallel iterated RKN methods over sequencial DIRKN methods from the literature is demonstrated.