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Uniformly accurate multiscale time integrators for second order oscillatory differential equations with large initial data

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Abstract

We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done.

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Acknowledgements

This work is supported partially by the French ANR project MOONRISE ANR-14-CE23-0007-01 and partially by the Singapore A*STAR SERC PSF-Grant 1321202067. Part of the work was done the author was visiting the Institute for Mathematical Sciences at the National University of Singapore in 2015. The author would like to thank Prof. Weizhu Bao for stimulating discussion and thank the referees for their valuable suggestions that greatly improves the paper.

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  1. IRMAR, Université de Rennes 1, Rennes, France

    Xiaofei Zhao

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  1. Xiaofei Zhao
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Correspondence to Xiaofei Zhao.

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Communicated by David Cohen.

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Zhao, X. Uniformly accurate multiscale time integrators for second order oscillatory differential equations with large initial data. Bit Numer Math 57, 649–683 (2017). https://doi.org/10.1007/s10543-017-0646-0

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