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Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs

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Abstract

In the last few decades, Runge-Kutta-Nyström (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems.

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Acknowledgments

We are grateful to Professor Christian Lubich for his careful reading and helpful comments on this manuscript. This work was done in part in Tübingen when the first author visited to UNIVERSITÄT TÜBINGEN in 2108.

Funding

The research is supported in part by the National Natural Science Foundation of China under Grants 11671200, 11401333, 11801377 and by Natural Science Foundation of Shandong Province under Grant ZR2014AQ003.

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Correspondence to Xinyuan Wu.

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Wu, X., Wang, B. & Mei, L. Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs. Numer Algor 86, 693–727 (2021). https://doi.org/10.1007/s11075-020-00908-7

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