High Order Schemes for Reaction-Diffusion Singularly Perturbed Systems

Abstract

In this paper we are interested in solving efficiently a singularly perturbed linear system of differential equations of reaction-diffusion type. Firstly, a non-monotone finite difference scheme of HODIE type is constructed on a Shishkinmesh. The previous method is modified at the transition points such that an inverse monotone scheme is obtained.We prove that if the diffusion parameters are equal it is a third order uniformly convergent method. If the diffusion parameters are different some numerical evidence is presented to suggest that an uniformly convergent scheme of order greater than two is obtained. Nevertheless, the uniform errors are bigger and the orders of uniform convergence are less than in the case corresponding to equal diffusion parameters.