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A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach

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Abstract

In this paper, we present a linearly implicit energy-preserving scheme for the Camassa–Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradient systems. The Camassa–Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank–Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme.

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Acknowledgements

The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. Chaolong Jiang’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11901513), the Yunnan Provincial Department of Education Science Research Fund Project (Grant No. 2019J0956) and the Science and Technology Innovation Team on Applied Mathematics in Universities of Yunnan. Yuezheng Gong’s work is partially supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20180413), the National Natural Science Foundation of China (Grant No. 11801269) and the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (Grant No. 202002). Wenjun Cai’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11971242) and the National Key Research and Development Project of China (Grant Nos. 2018YFC0603500, 2018YFC1504205). Yushun Wang’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11771213).

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Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, People’s Republic of China

    Chaolong Jiang

  2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Yuezheng Gong

  3. Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Wenjun Cai & Yushun Wang

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  1. Chaolong Jiang
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  2. Yuezheng Gong
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  3. Wenjun Cai
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  4. Yushun Wang
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Correspondence to Yushun Wang.

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Jiang, C., Gong, Y., Cai, W. et al. A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach. J Sci Comput 83, 20 (2020). https://doi.org/10.1007/s10915-020-01201-4

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  • DOI: https://doi.org/10.1007/s10915-020-01201-4

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