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

TL;DR

This paper introduces a second-order shifted BDF2 scheme for solving the 2D Fisher-KPP equation, demonstrating stability and accuracy with both uniform and non-uniform time steps through numerical experiments.

Contribution

It proposes a novel second-order shifted BDF2 method for the 2D Fisher-KPP equation, including stability analysis and validation with numerical experiments.

Findings

The scheme is stable for uniform time steps.

Numerical experiments confirm robustness and accuracy.

Both uniform and non-uniform time stepping are effective.

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.