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Efficient Classes Of Runge-Kutta Methods For Two-point Boundary Value Problems

W. Enright, P. Muir
Published 2005 · Computer Science, Mathematics

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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.ZusammenfassungDie standardmäßige Verwendung von impliziten RK-Verfahren zur Lösung von 2-Punkt Randwertproblemen erfordert für die Berechnung der Stufen die Lösung eines Systems vonn×s nichtlinearen Gleichungen, won die Anzahl der Differentialgleichungen unds die Stufenanzahl des impliziten RK-Verfahrens ist. Man kann jedoch für 2-Punkt Randwertprobleme eine Teilmenge der IRK-Verfahren auswählen, für die die Lösung von Gleichungssystemen nicht nötig ist; die Stufen können, wie bei expliziten RK-Verfahren, direkt berechnet werden. Trotzdem haben diese Verfahren bessere Stabilitätseigenschaften als explizite RK-Verfahren. Wir nennen diese neuen Formeln 2-Punkt explizite RK (TPERK)-Verfahren. Weil ihre Stufen explizit berechnet werden können, kann mit ihnen die Lösung eines 2-Punkt-Randwertproblems effizienter als mit einem impliziten RK-Verfahren berechnet werden. Wir beschreiben auch eine Unterklasse von symmetrischen TPERK-Verfahren, die ATPERK-Verfahren, die eine Reihe nützlicher Eigenschaften aufweisen.
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