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Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

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  • Almushaira, Mustafa

Abstract

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

Suggested Citation

  • Almushaira, Mustafa, 2023. "Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    • Handle: RePEc:eee:apmaco:v:451:y:2023:i:c:s0096300323002084
    DOI: 10.1016/j.amc.2023.128039
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    References listed on IDEAS

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    1. Jiang, Chaolong & Sun, Jianqiang & Li, Haochen & Wang, Yifan, 2017. "A fourth-order AVF method for the numerical integration of sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 144-158.
    2. Xing, Zhiyong & Wen, Liping & Wang, Wansheng, 2021. "An explicit fourth-order energy-preserving difference scheme for the Riesz space-fractional Sine–Gordon equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 624-641.
    3. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
    4. Sheng, Q. & Khaliq, A.Q. M. & Voss, D.A., 2005. "Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(4), pages 355-373.
    5. Almushaira, M. & Bhatt, H. & Al-rassas, A.M., 2021. "Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 235-258.
    6. Hu, Dongdong & Cai, Wenjun & Xu, Zhuangzhi & Bo, Yonghui & Wang, Yushun, 2021. "Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine–Gordon equation with damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 35-59.
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