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Parallel diagonally implicit Runge-Kutta-Nyström methods

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. © 1992.

 van der Houwen P.J., Sommeijer B.P., Cong N.h.
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  Từ khóa : Mathematical Models; DIRKN Methods; High Stage Order; Lipschitz Constants; Parallel Computers; RKN Methods; Runge-Kutta-Nystroem Methods; Mathematical Techniques