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The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations

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  • Deng, Dingwen
  • Liang, Dong

Abstract

Nonlinear wave equation is extensively applied in a wide variety of scientific fields, such as nonlinear optics, solid state physics and quantum field theory. In this paper, two high-performance compact alternating direction implicit (ADI) methods are developed for the nonlinear wave equations. The first scheme is developed a three-level nonlinear difference scheme for nonlinear wave equations, where in x-direction, series of linear tridiagonal systems are solved by Thomas algorithm, while in y-direction, nonlinear algebraic system are computed by Newton’s iterative method. In contrast, the second scheme is linear, and permits the multiple uses of the Thomas algorithm in both x- and y-directions, thus it saves much time cost. By using the discrete energy analysis method, it is shown that both the developed schemes can attain numerical accuracy of order O(τ4+hx4+hy4) in H1-norm. Meanwhile, by the fixed point theorem and symmetric positive-definite properties of coefficient matrix, it is proved that they are both uniquely solvable. Besides, the proposed schemes are extended to the numerical solutions of the coupled sine-Gordon wave equations and damped wave equations. Finally, numerical results confirm the convergence orders and exhibit efficiency of our algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:329:y:2018:i:c:p:188-209
    DOI: 10.1016/j.amc.2018.02.010
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    References listed on IDEAS

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    1. Wazwaz, Abdul-Majid, 2006. "Compactons, solitons and periodic solutions for some forms of nonlinear Klein–Gordon equations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1005-1013.
    2. A. Mayo, 2004. "High-order accurate implicit finite difference method for evaluating American options," The European Journal of Finance, Taylor & Francis Journals, vol. 10(3), pages 212-237.
    3. Sun, Yunchuan, 2015. "New exact traveling wave solutions for double Sine–Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 100-104.
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    Cited by:

    1. Xie, Jianqiang & Zhang, Zhiyue, 2019. "An analysis of implicit conservative difference solver for fractional Klein–Gordon–Zakharov system," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 153-166.
    2. Xie, Jianqiang & Wang, Quanxiang & Zhang, Zhiyue, 2023. "Conservative finite difference methods for the Boussinesq paradigm equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 588-613.
    3. 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.

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