On symmetric Runge-Kutta methods of high order

The usual characterization of symmetry for Runge-Kutta methods is that given by Stetter. In this paper an equivalent characterization of symmetry based on theW-transformation of Hairer and Wanner is proposed. Using this characterization it is simple to show symmetry for some well-known classes of high order Runge-Kutta methods which are based on quadrature formulae. It can also be used to construct a one-parameter family of symmetric and algebraically stable Runge-Kutta methods based on Lobatto quadrature. Methods constructed in this way and presented in this paper extend the known class of implicit Runge-Kutta methods of high order.ZusammenfassungDie übliche Charakterisierung der Symmetrie für Runge-Kutta Methoden ist die von Stetter angegebene. In dieser Arbeit wird eine äquivalente Charakterisierung vorgeschlagen, die auf derW-Transformation von Hairer und Wanner beruht. Mit dieser Charakterisierung kann die Symmetrie für einige Klassen von Runge-Kutta Methoden einfach gezeigt werden. Sie kann auch dazu benützt werden, eine einparametrige Familie von symmetrischen und algebraisch stabilen Runge-Kutta Methoden, die auf der Lobatto-Quadratur beruhen, zu konstruieren. Damit kann die Klasse impliziter Runge-Kutta Methoden höherer Ordnung erweitert werden.

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