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Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

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Abstract

It is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semi-linear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. The results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First, a multi-frequency modulated Fourier expansion of the AAVF method is constructed, and then two almost-invariants of the modulation system are derived.

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Acknowledgements

The authors sincerely thank the two anonymous reviewers for their valuable suggestions, which helped improve this paper significantly. The authors are grateful to Professor Christian Lubich for his helpful comments and discussions on the topic of modulated Fourier expansions. We also thank him for drawing our attention to the long-term analysis of energy-preserving methods, which motives this paper.

Funding

The research of the first author is financially supported in part by the Alexander von Humboldt Foundation and by the Natural Science Foundation of Shandong Province (Outstanding Youth Foundation) under Grant ZR2017JL003. The research of the second author is financially supported in part by the National Natural Science Foundation of China under Grant 11671200.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Bin Wang

  2. Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Bin Wang

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xinyuan Wu

  4. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Xinyuan Wu

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  1. Bin Wang
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  2. Xinyuan Wu
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Correspondence to Bin Wang.

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Communicated by: Enrique Zuazua

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Wang, B., Wu, X. Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations. Adv Comput Math 45, 2921–2952 (2019). https://doi.org/10.1007/s10444-019-09719-4

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  • DOI: https://doi.org/10.1007/s10444-019-09719-4

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