Solving efficiently one dimensional parabolic singularly perturbed reaction-diffusion systems: A splitting by components

In this paper we consider 1D parabolic singularly perturbed systems of reaction-diffusion type which are coupled in the reaction term. The numerical scheme, used to approximate the exact solution, combines the fractional implicit Euler method and a splitting by components to discretize in time, and the classical central finite differences scheme to discretize in space. The use of the fractional Euler method combined with the splitting by components means that only tridiagonal linear systems must be solved to obtain the numerical solution. For simplicity, the analysis is presented in a complete form only in the case of systems which have two equations, but it can be easily extended to an arbitrary number of equations. If a special nonuniform mesh in space is used, the method is uniformly and unconditionally convergent, having first order in time and almost second order in space. Some numerical results are shown which corroborate in practice the theoretical ones.