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

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.