High-order Compact Difference Methods for Simulating Wave Propagations in Excitable Media

In this paper, we present a study of some high-order compact difference schemes for solving the Fitzhugh-Nagumo equations governed by two coupled time-dependent nonlinear reaction diffusion equations in two variables. Solving the Fitzhugh-Nagumo equations is quite challenging, since the equations involve spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed have sixth order accuracy in space, and fourth order in time if the fourth order Runge-Kutta method is adopted for time marching. To improve efficiency, we also propose an ADI scheme (for two dimensional problems), which has second order accuracy in time. Numerical results are presented for plane wave propagation in one dimension and spiral waves for two dimensions.