Numerical methods for stiff reaction-diffusion systems

In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their accuracy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time steps. IIF2 is found to be more accurate for systems with large diffusion while CN-MG is more efficient when the number of spatial grid points is large.