A sixth-order compact finite difference framework for solving nonlinear reaction-diffusion equations: application to FitzHugh-Nagumo model

Jiang Fu; Xiao-Yu Zhang; Qing Fang; Jiang Fu; Xiao-Yu Zhang; Qing Fang

doi:10.3934/math.2025940

AIMS Mathematics

2025, Volume 10, Issue 9: 21040-21060. doi: 10.3934/math.2025940

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A sixth-order compact finite difference framework for solving nonlinear reaction-diffusion equations: application to FitzHugh-Nagumo model

1.
Department of Mathematics, College of Science, Beijing Forestry University, Beijing, 100083, China
2.
Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata, 990-8560, Japan

Received: 18 June 2025 Revised: 28 August 2025 Accepted: 01 September 2025 Published: 12 September 2025
MSC : 35K57, 65M06, 65M12

This paper proposes a sixth-order compact finite difference framework to numerically solve nonlinear reaction-diffusion equations, with a particular focus on the FitzHugh-Nagumo (FHN) model. First, for the second-order spatial derivatives in the FHN equation, a five-point sixth-order compact difference scheme is used for internal points, and a asymmetric six-point compact difference scheme is used for boundary points to achieve spatial discretization, thereby transforming the problem into an ordinary differential equation; then, this is and then combined with the semi-implicit Crank-Nicholson method for the time discretization to obtain a numerical solution scheme for the FHN equation. We establish the stability and convergence of the method and validate it through numerical experiments. The feasibility and accuracy of the method were verified by conducting an error analysis on the numerical results and comparing them with other algorithms. It is proven that this method is an effective tool to solve the numerical solutions of nonlinear reaction-diffusion equations.
- FitzHugh-Nagumo equation,
- nonlinear reaction-diffusion equation,
- numerical method,
- compact finite difference method,
- Crank-Nicolson method
Citation: Jiang Fu, Xiao-Yu Zhang, Qing Fang. A sixth-order compact finite difference framework for solving nonlinear reaction-diffusion equations: application to FitzHugh-Nagumo model[J]. AIMS Mathematics, 2025, 10(9): 21040-21060. doi: 10.3934/math.2025940

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Abstract

This paper proposes a sixth-order compact finite difference framework to numerically solve nonlinear reaction-diffusion equations, with a particular focus on the FitzHugh-Nagumo (FHN) model. First, for the second-order spatial derivatives in the FHN equation, a five-point sixth-order compact difference scheme is used for internal points, and a asymmetric six-point compact difference scheme is used for boundary points to achieve spatial discretization, thereby transforming the problem into an ordinary differential equation; then, this is and then combined with the semi-implicit Crank-Nicholson method for the time discretization to obtain a numerical solution scheme for the FHN equation. We establish the stability and convergence of the method and validate it through numerical experiments. The feasibility and accuracy of the method were verified by conducting an error analysis on the numerical results and comparing them with other algorithms. It is proven that this method is an effective tool to solve the numerical solutions of nonlinear reaction-diffusion equations.

References

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