An Unconditionally Stable Positivity-Preserving Scheme for the One-Dimensional Fisher–Kolmogorov–Petrovsky–Piskunov Equation

Sangkwon Kim,Seungyoon Kang,Soobin Kwak,Chaeyoung Lee,Seokjun Ham,Junseok Kim,Hyundong Kim,Youngjin Hwang,Hyun Geun Lee,Youssef N Raffoul

doi:10.1155/2021/7300471

Sangkwon Kim, Seungyoon Kang + Show 8 more

Open Access

https://doi.org/10.1155/2021/7300471

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Abstract

In this study, we present an unconditionally stable positivity-preserving numerical method for the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation in the one-dimensional space. The Fisher–KPP equation is a reaction-diffusion system that can be used to model population growth and wave propagation. The proposed method is based on the operator splitting method and an interpolation method. We perform several characteristic numerical experiments. The computational results demonstrate the unconditional stability, boundedness, and positivity-preserving properties of the proposed scheme.

Highlights

In an infinitely long domain, the behavior of virile mutant propagation can be modeled by Fisher’s equation [1]. It was independently studied by Kolmogorov, Petrovsky, and Piskunov [2]. ere are some numerical methods to solve the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) or fractional Fisher–KPP equations such as the spectral collocation-type method [3, 4], Adams–Bashforth–Moulton method [5], and Mittag-Leffler kernel [6]
We present an unconditionally stable positivity-preserving numerical method for the Fisher–KPP equation: zu z2u zt (x, t) D z2x (x, t) + Ku(x, t) up(x, t)􏼁, (1)
If p in equation (1) is large, the Fisher–KPP equation becomes a very stiff problem to be numerically solved. To resolve this stiff problem, we propose the operator splitting method (OSM) with a recently developed interpolation method [29]. e computational results demonstrate the unconditional stability and positivity-preserving properties of the proposed scheme

Summary

Introduction

In an infinitely long domain, the behavior of virile mutant propagation can be modeled by Fisher’s equation [1]. We present an unconditionally stable positivity-preserving numerical method for the Fisher–KPP equation: zu z2u zt (x, t) D z2x (x, t) + Ku(x, t). Ilati and Dehghan [19] proposed the local boundary integral equation method to solve the extended Fisher–KPP equation. Numerical research studies for the nonlocal Fisher–KPP equation have been actively conducted to validate the asymptotic behavior [24], positivity of the solution [25, 26], and stability of the travelling wave solution [27, 28]. If p in equation (1) is large, the Fisher–KPP equation becomes a very stiff problem to be numerically solved To resolve this stiff problem, we propose the operator splitting method (OSM) with a recently developed interpolation method [29].

Numerical Solution Algorithm

Numerical Experiments

Conclusions