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Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme

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  • Poochinapan, Kanyuta
  • Wongsaijai, Ben

Abstract

In this paper, we present a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The proposed scheme is described by the compact difference operators together with the additional stabilized term. As a matter of fact, the Allen-Cahn equation contains the nonlinear reaction term which is eminently proved that numerical schemes are mostly nonlinear. To solve the complexity of nonlinearity, the Crank-Nicolson/Adams-Bashforth method is applied in order to deal with the nonlinear terms with the linear implicit scheme. The well-known energy-decaying property of the equation is maintained by the proposed scheme in the discrete sense. Additionally, the L∞ error analysis is carried out in the 1D case in a rigorous way to show that the method is fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Concurrently, we examine the L2 and H1 error analysis for the scheme in the case of 2D. We consider the impact of the additional stabilized term on numerical solutions. The consequences confirm that an appropriate value of the stabilized term yields a significant improvement. Moreover, relevant results are carried out in the numerical simulations to illustrate the faithfulness of the present method by the confirmation of existing pieces of evidence.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:434:y:2022:i:c:s0096300322004489
    DOI: 10.1016/j.amc.2022.127374
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    References listed on IDEAS

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    1. Wongsaijai, B. & Oonariya, C. & Poochinapan, K., 2020. "Compact structure-preserving algorithm with high accuracy extended to the improved Boussinesq equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 125-150.
    2. Choi, Jeong-Whan & Lee, Hyun Geun & Jeong, Darae & Kim, Junseok, 2009. "An unconditionally gradient stable numerical method for solving the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(9), pages 1791-1803.
    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. Prakash, Amit & Kaur, Hardish, 2019. "Analysis and numerical simulation of fractional order Cahn–Allen model with Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 134-142.
    5. Rouatbi, Asma & Omrani, Khaled, 2017. "Two conservative difference schemes for a model of nonlinear dispersive equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 516-530.
    6. 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.
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    Cited by:

    1. Uzunca, Murat & Karasözen, Bülent, 2023. "Linearly implicit methods for Allen-Cahn equation," Applied Mathematics and Computation, Elsevier, vol. 450(C).

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