Efficient classes of Runge-Kutta methods for two-point boundary value problems |
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Authors: | W. H. Enright P. H. Muir |
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Affiliation: | 1. Dept. of Computer Science, University of Toronto, MSS 1A4, Toronto, Ontario, Canada 2. Dept. of Mathematics and Computer Science, Saint Mary's University, B3H 3C3, Halifax, Nova Scotia, Canada
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Abstract: | The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties. |
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