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Numerical Conservations of Energy, Momentum and Actions in the Full Discretisation for Nonlinear Wave Equations

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Abstract

This paper analyses the long-time behaviour of one-stage symplectic or symmetric trigonometric integrators when applied to nonlinear wave equations. It is shown that energy, momentum, and all harmonic actions are approximately preserved over a long time for one-stage explicit symplectic or symmetric trigonometric integrators when applied to nonlinear wave equations via spectral semi-discretisations. For the long-term analysis of symplectic or symmetric trigonometric integrators, we derive a multi-frequency modulated Fourier expansion of the trigonometric integrator and show three almost-invariants of the modulation system. In the analysis of this paper, we neither assume symmetry for symplectic methods, nor assume symplecticity for symmetric methods. The results for symplectic and symmetric methods are obtained as a byproduct of the above analysis. We also give another proof by establishing a relationship between symplectic and symmetric trigonometric integrators and trigonometric integrators which have been researched for wave equations in the literature.

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Data Availibility Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors sincerely thank the anonymous reviewers for the very valuable comments and helpful suggestions. This work was supported by NSFC (12371403, 12271426), Key Research and Development Projects of Shaanxi Province (2023-YBSF-399) and China Postdoctoral Science Foundation (2023M732869).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Zhen Miao

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Bin Wang & Yao-Lin Jiang

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  1. Zhen Miao
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  2. Bin Wang
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  3. Yao-Lin Jiang
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Correspondence to Bin Wang.

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Miao, Z., Wang, B. & Jiang, YL. Numerical Conservations of Energy, Momentum and Actions in the Full Discretisation for Nonlinear Wave Equations. J Sci Comput 98, 10 (2024). https://doi.org/10.1007/s10915-023-02405-0

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  • DOI: https://doi.org/10.1007/s10915-023-02405-0

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