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.

KEYWORDS:

• Perturbed oscillatory systems
• Hamiltonian systems
• multiderivative extended Runge–Kutta–Nyström methods
• order conditions
• phase properties

2010 AMS SUBJECT CLASSIFICATION:

• 65L20

Acknowledgements

The authors are greatly indebted to the referees for useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (11771112, 11671112).