Approximating Runge-Kutta matrices by triangular matrices |
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Authors: | W. Hoffmann J. J. B. De Swart |
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Affiliation: | (1) Department of Mathematics and Computer Science, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands;(2) Department of Numerical Mathematics, CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands |
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Abstract: | The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences). |
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Keywords: | Primary 65L06 Secondary 15A23 |
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