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Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations

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Abstract

A key issue in developing efficient numerical schemes for nonlinear wave equations is the energy-conserving. Most existing schemes of the energy-conserving are fully implicit and the schemes require an extra iteration at each time step and considerable computational cost for a long time simulation, while the widely-used q-stage (implicit) Gauss scheme (method) only preserves polynomial Hamiltonians up to degree 2q. In this paper, we present a family of linearly implicit and high-order energy-conserving schemes for solving nonlinear wave equations. The construction of schemes is based on recently-developed scalar auxiliary variable technique with a combination of classical high-order Gauss methods and extrapolation approximation. We prove that the proposed schemes are unconditionally energy-conserved for a general nonlinear wave equation. Numerical results are given to show the energy-conserving and the effectiveness of schemes.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  3. Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, Zhuhai, 519087, China

    Weiwei Sun

  4. Division of Science and Technology, United International College (BNU-HKBU), Zhuhai, 519087, China

    Weiwei Sun

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  1. Dongfang Li
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  2. Weiwei Sun
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Correspondence to Dongfang Li.

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This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11771162, 11971010 and a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project CityU 11302519).

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Li, D., Sun, W. Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations. J Sci Comput 83, 65 (2020). https://doi.org/10.1007/s10915-020-01245-6

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