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Explicit, non-negativity-preserving and maximum-principle-satisfying finite difference scheme for the nonlinear Fisher's equation

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  • Deng, Dingwen
  • Xiong, Xiaohong

Abstract

In this paper, a class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O(Δt+(Δt/hx)2+hx2) for one-dimensional case and O(Δt+(Δt/hx)2+(Δt/hy)2+hx2+hy2) for two-dimensional case in the maximum norm, respectively. Here, Δt, hx and hy are meshsizes in t-, x- and y-directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems.

Suggested Citation

  • Deng, Dingwen & Xiong, Xiaohong, 2024. "Explicit, non-negativity-preserving and maximum-principle-satisfying finite difference scheme for the nonlinear Fisher's equation," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    • Handle: RePEc:eee:apmaco:v:466:y:2024:i:c:s0096300323006367
    DOI: 10.1016/j.amc.2023.128467
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    References listed on IDEAS

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    1. Dag, Idiris & Ersoy, Ozlem, 2016. "The exponential cubic B-spline algorithm for Fisher equation," Chaos, Solitons & Fractals, Elsevier, vol. 86(C), pages 101-106.
    2. Feng, Zhaosheng & Li, Yang, 2006. "Complex traveling wave solutions to the Fisher equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 115-123.
    3. Tan, Yue & Xu, Hang & Liao, Shi-Jun, 2007. "Explicit series solution of travelling waves with a front of Fisher equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 462-472.
    4. Balyan, L.K. & Mittal, A.K. & Kumar, M. & Choube, M., 2020. "Stability analysis and highly accurate numerical approximation of Fisher’s equations using pseudospectral method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 86-104.
    5. Qin, Wendi & Ding, Deqiong & Ding, Xiaohua, 2015. "Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 552-567.
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