Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.