A Fourth-Order Compact Finite Difference Scheme to the Numerical Solution of FitzHugh-Nagumo Equation

Abstract

The FitzHugh-Nagumo equation is an important nonlinear reaction-diffusion equation used in physics and chemicals. To obtain the numerical solution of partial differential equations, the compact finite difference method is widely applied. In this paper, I propose a new numerical solution to FitzHugh-Nagumo equation by using a fourth-order compact finite difference scheme in space, and a semi-implicit Crank-Nicholson method in time. I further calculate the results in terms of accuracy by leveraging the proposed method and exact solution. In particular, I compare the new method whose convergence order is close to four with the second order central difference method. The simulated results show the new solution is more accurate and effective. The proposed method is expected to be a good solution to some problems in the real world.