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Study on the dynamics of a nonlinear dispersion model in both 1D and 2D based on the fourth-order compact conservative difference scheme

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  • Dimitrienko, Yu.I.
  • Li, Shuguang
  • Niu, Yi

Abstract

In this paper, the nonlinear dispersion wave model in both 1D and 2D is studied by the compact finite difference method, which is called the generalized Rosenau–RLW equation. A fourth-order compact three-level and linearized difference scheme that maintains the original conservative properties of equation is proposed. The discrete mass conservation and discrete energy conservation of compact difference scheme are obtained. The solvability of numerical scheme is obtained. By using the discrete energy method, convergence and unconditional stability can also be obtained without relying on the grid ratio, and the optimal error estimates in the L∞ norm are fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. The scheme is conservative so can be used for long time computation. Numerical experiment results show that the theory is accurate and the method is efficient and reliable. Finally, the new numerical scheme is used to study the nonlinear dynamic of 1D generalized Rosenau–RLW equation and the wave interference of 2D generalized Rosenau–RLW equation, focusing on the influence of nonlinear convection term on the wave and the conservation of wave propagation.

Suggested Citation

  • Dimitrienko, Yu.I. & Li, Shuguang & Niu, Yi, 2021. "Study on the dynamics of a nonlinear dispersion model in both 1D and 2D based on the fourth-order compact conservative difference scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 661-689.
    • Handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:661-689
    DOI: 10.1016/j.matcom.2020.11.012
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    References listed on IDEAS

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    1. Jinsong Hu & Yulan Wang, 2013. "A High-Accuracy Linear Conservative Difference Scheme for Rosenau-RLW Equation," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, November.
    2. Wang, Xiaofeng & Dai, Weizhong & Guo, Shuangbing, 2019. "A conservative linear difference scheme for the 2D regularized long-wave equation," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 55-70.
    3. Wongsaijai, B. & Mouktonglang, T. & Sukantamala, N. & Poochinapan, K., 2019. "Compact structure-preserving approach to solitary wave in shallow water modeled by the Rosenau-RLW equation," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 84-100.
    4. Xintian Pan & Luming Zhang, 2012. "Numerical Simulation for General Rosenau-RLW Equation: An Average Linearized Conservative Scheme," Mathematical Problems in Engineering, Hindawi, vol. 2012, pages 1-15, May.
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    Cited by:

    1. Mouktonglang, Thanasak & Yimnet, Suriyon & Sukantamala, Nattakorn & Wongsaijai, Ben, 2022. "Dynamical behaviors of the solution to a periodic initial–boundary value problem of the generalized Rosenau-RLW-Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 114-136.
    2. Poochinapan, Kanyuta & Wongsaijai, Ben, 2023. "High-performance computing of structure-preserving algorithm for the coupled BBM system formulated by weighted compact difference operators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 439-467.
    3. Poochinapan, Kanyuta & Wongsaijai, Ben, 2022. "Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme," Applied Mathematics and Computation, Elsevier, vol. 434(C).

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