Explicit Runge-Kutta methods for initial value problems with oscillating solutions

Abstract

New pairs of embedded Runge-Kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained. These pairs have been derived taking into account not only the usual properties of accuracy, stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by Van der Houwen and Sommeijer (1989). Three nine-stage embedded pairs of Runge-Kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and/or dissipation are selected among the members of a family of pairs depending on several free parameters. Some numerical results are presented to show the efficiency of the new methods.