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Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations

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  • Deng, Dingwen
  • Wu, Qiang

Abstract

The nonlinear couple wave equations, which are extensively applied in scientific fields, such as, solid state physics, quantum mechanics, nonlinear optics, are a kind of important evolution equations. This paper is concerned with their numerical solutions via the combinations of compact difference method, Predictor–Corrector (P–C) iterative methods and Richardson extrapolation methods (REMs). Firstly, fourth-order compact difference methods are used to discrete temporal and spatial derivatives, thus forming a nonlinear fully discrete compact difference scheme. By utilizing the discrete energy analysis method and fixed point theorem, we can prove that under the condition of accepted stable criterion this scheme is conditionally convergent with an order of O(τ4+hx4+hy4) in H1-norm, and solvable. For avoiding solving the system of nonlinear algebraic equations, a P–C iterative method is introduced to save time cost and make the implementation simple. Besides, REMs are further applied to attain higher-order accurate approximate solutions. Finally, numerical findings confirm the exactness of theoretical results and efficiency of the algorithms.

Suggested Citation

  • Deng, Dingwen & Wu, Qiang, 2023. "Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 223-249.
    • Handle: RePEc:eee:matcom:v:203:y:2023:i:c:p:223-249
    DOI: 10.1016/j.matcom.2022.06.030
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    References listed on IDEAS

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    1. Kumar, Manoj & Daftardar-Gejji, Varsha, 2019. "A new family of predictor-corrector methods for solving fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    2. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    3. Yun-Mei Zhao, 2014. "Exact Solutions of Coupled Sine-Gordon Equations Using the Simplest Equation Method," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-5, February.
    4. Deng, Dingwen & Liang, Dong, 2018. "The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 188-209.
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