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Multiderivative extended Runge–Kutta–Nyström methods for multi-frequency oscillatory systems

ABSTRACT In this paper, a class of multistage and multivalue numerical methods called multiderivative extended Runge–Kutta–Nyström (MERKN) methods are presented for the integration of multi-frequency oscillatory systems. The order conditions of these schemes are derived by using the theory of extended Nyström trees and B-series. To simplify the construction of the new methods, the order conditions are simplified by omitting the redundant trees. Based on the simplified-order conditions, a family of three-stage and one-parameter explicit MERKN methods of order five is constructed. Meanwhile, the order of energy preservation, the stability and the phase properties of the new methods are analysed, and a three-stage method with minimal dispersion error is obtained. The results of numerical experiments demonstrate the efficiency of our new methods in comparison with some other RKN-type methods.