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The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

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Abstract

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of \({\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})\) in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

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Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions, which have helped to improve the paper. We are also very grateful to Editor in Chief Claude Brezinski for his kind help.

Funding

This work is partly supported by the National Natural Science Foundation of China (Nos. 11861047, 11871393), Natural Science Foundation of Jiangxi province (No. 20202BABL201005), and a key project of the International Science and Technology Cooperation Program of Shaanxi Research and Development Plan (Grant no. 2019KWZ-08).

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  1. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, China

    Dingwen Deng & Qiang Wu

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  1. Dingwen Deng
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  2. Qiang Wu
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Correspondence to Dingwen Deng.

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Deng, D., Wu, Q. The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations. Numer Algor 88, 1875–1914 (2021). https://doi.org/10.1007/s11075-021-01099-5

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  • DOI: https://doi.org/10.1007/s11075-021-01099-5

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