A comparison of two numerical methods for oscillatory reaction-diffusion systems

Reaction-diffusion systems whose kinetics contain a stable limit cycle are an established class of models for a range of oscillatory biological and chemical phenomena. In this paper, the author compares two numerical methods for calculating the oscillatory wake solutions generated by spatially localized perturbations for one particular reaction-diffusion system, of lambda-omega type. The two methods are a semi-implicit, or implicit-explicit, finite difference scheme based on the Crank-Nicolson algorithm, and the method of lines with Gear's method. Though both solutions ultimately converge to a common solution, the approach to this final solution is very different in the two cases. The results provide a clear illustration of the care required in numerical solution of oscillatory reaction-diffusion equations