A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

Abstract

In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.

Keywords: Reaction-diffusion systems, singular perturbations, fitted operator finite
difference methods, convergence analysis.