A second-order generalized BDF method for the two-dimensional (modified) Fisher-Kolmogorov-Petrovsky-Piskunov equation

2025/07/20 by Lei Ge, Ge, Lei, Yongliang Zhao +3
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2507.14939

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) equation is a classical reaction-diffusion equation with broad applications such as biology, chemistry and physics. In this paper, an alternative second-order scheme is proposed by employing a shifted BDF2 method to approximate the two-dimensional (modified) Fisher-KPP equation. We both consider an uniform and a nonuniform time steps of such the scheme. The stability of the uniform discretization scheme is proved. Numerical experiments demonstrate that our uniform and non-uniform schemes are robust and accurate.

Citations

• On a new class of BDF and IMEX schemes for parabolic type equations
• Structure-preserving weighted BDF2 methods for Anisotropic Cahn-Hilliard model: uniform/variable-time-steps
• Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations
• Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds
• Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations
• Multidimensional nonlinear diffusion arising in population genetics
• The chemical basis of morphogenesis

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