Several Finite Difference Methods For FitzHugh-Nagumo Equation

Fitz Hugh-Nagumo(FHN) equations are semilinear reaction diffusion equations. It is an important model in analyzing the current transportation in nerve, and it has wide application in many fields such as chemistry, physics and metallurgy. In literature, there are many theoretical results but few works on numerical approximation of FHN equations. In this paper, several effective numerical schemes are put forward. The main contributions of this work are as follows:(1) Implicit finite difference scheme and Crank-Nicolson scheme are used to solve Fitz Hugh-Nagumo equations. By linearizing the nonlinear reaction terms in semi-implicit scheme, we get semi-implicit finite difference scheme to the nonlinear equations. The scheme has not only good stability but good computation efficiency. We get three numerical approaches for each of two schemes respectively. Comparison studies of different schemes on the errors and convergence rate are conducted. The numerical experiments show the effectiveness of the schemes.(2) A weighted numerical scheme has been proposed. And the same linearization approach is adopted to deal with the nonlinear term. The coefficients matrix of the scheme is an M-matrix, hence it is more stable. Numerical experiments are provided to verify the theoretical results.(3) With these schemes, when applied with different parameters, the numerical solutions are observed to have the same metastability property as Allen-Cahn equations. Even though it has not got theoretical support, it is supposed to be a field of interest.