Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lei Ge, Yong-Liang Zhao, Qian-Yu Shu

TL;DR

The paper tackles numerical solution of the two-dimensional (modified) Fisher–KPP equation by introducing a second-order GBDF2-IMEX scheme that is stable on uniform time steps and extendable to nonuniform grids. Spatial discretization uses second-order central differences, producing a Kronecker-structured linear system that benefits from a fast DST-based solver. The authors prove second-order convergence and stability for the uniform scheme and demonstrate comparable temporal and spatial accuracy on nonuniform grids through theoretical discussion and numerical experiments. Four test problems verify robustness across varying nonlinear terms, boundary conditions, and parameter choices, highlighting the method’s practicality for multiscale reaction-diffusion dynamics.

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.

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

TL;DR

The paper tackles numerical solution of the two-dimensional (modified) Fisher–KPP equation by introducing a second-order GBDF2-IMEX scheme that is stable on uniform time steps and extendable to nonuniform grids. Spatial discretization uses second-order central differences, producing a Kronecker-structured linear system that benefits from a fast DST-based solver. The authors prove second-order convergence and stability for the uniform scheme and demonstrate comparable temporal and spatial accuracy on nonuniform grids through theoretical discussion and numerical experiments. Four test problems verify robustness across varying nonlinear terms, boundary conditions, and parameter choices, highlighting the method’s practicality for multiscale reaction-diffusion dynamics.

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.

Paper Structure

This paper contains 7 sections, 2 theorems, 93 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Uniform GBDF2-IMEX scheme for the Fisher-KPP equation
  3. Discretization of the Fisher-KPP
  4. Stability and convergence of the uniform GBDF2-IMEX scheme
  5. Nonuniform GBDF2-IMEX discretization scheme
  6. Numerical experiments
  7. Concluding remarks

Key Result

Theorem 2.1

Assume Eq. eq2.2 and the solution of Eq. eq1.1 is sufficiently smooth such that Eq. eq2.11 is true, and the following stability condition is satisfied where $\rho$ is a constant. Given $u^{0}=u(0)\in V$, we assume $u^{1}$ is computed with a proper initialization procedure such that Then, for $\Delta t$ sufficiently small and $\beta>1$, we conclude that and where $g$ is a positive constant and

Figures (11)

  • Figure 1: Observed temporal errors for Example 1 for different values of $\beta$ and $M$, where $N=500$.
  • Figure 2: Observed space errors for Example 1 for different values of $\beta$ and $N$, where $M=1000$.
  • Figure 3: Observed temporal errors for Example 2 for different values of $\beta$ and $M$, where $N=500$.
  • Figure 4: Observed space errors for Example 2 for different values of $\beta$ and $N$, where $M=10000$.
  • Figure 5: Observed temporal errors for Example 3 for different values of $\beta$ and $M$, where $N=500$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Proof 1
  • Remark 2.2
  • Theorem 2.3
  • Proof 2
  • Remark 3.1