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Arbitrarily High-Order Energy-Preserving Schemes for the Zakharov-Rubenchik Equations

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Abstract

In this paper, we present a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations. The main idea of the scheme is first to introduce an quadratic auxiliary variable to transform the Hamiltonian energy into a modified quadratic energy and the original system is then reformulated into an equivalent system which satisfies the mass, modified energy as well as two linear invariants. The symplectic Runge-Kutta method in time, together with the Fourier pseudo-spectral method in space is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve arbitrarily high-order accurate in time and conserve the three invariants: mass, Hamiltonian energy and two linear invariants. In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are addressed to validate the theoretical results.

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Acknowledgements

The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. The work is supported by the National Natural Science Foundation of China (Grant Nos. 11901513, 12261097, 12261103), and the Yunnan Fundamental Research Projects (Grant Nos. 202101AT070208, 202001AT070066, 202101AS070044), the Natural Science Foundation of Hunan (Grant No. 2021JJ40655) and Innovation team of School of Mathematics and Statistics, Yunnan University (No. ST20210104). The first author is in particular grateful to Prof. Weizhu Bao for fruitful discussions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yunnan University, Kunming, 650504, China

    Gengen Zhang

  2. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China

    Chaolong Jiang & Hao Huang

  3. Department of Mathematics, College of Liberal Arts and Science, National University of Defense Technology, Changsha, 410073, China

    Chaolong Jiang

  4. Basic Teaching Department, Guizhou Vocational Institute of Technology, Fuquan, 550500, China

    Hao Huang

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  1. Gengen Zhang
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Correspondence to Chaolong Jiang.

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Zhang, G., Jiang, C. & Huang, H. Arbitrarily High-Order Energy-Preserving Schemes for the Zakharov-Rubenchik Equations. J Sci Comput 94, 32 (2023). https://doi.org/10.1007/s10915-022-02075-4

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