Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion problems

Abstract

In this paper, we present an optimal fourth-order parameter-uniform non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion boundary value problems exhibiting boundary layers at both ends of the domain. We discretize the problem using a high-order non-monotone finite difference scheme and prove that the scheme is stable in the maximum norm. The equidistribution of an appropriate monitor function is used to generate the layer-adapted meshes to discretize the problem. The method is proved to be optimal fourth-order uniformly convergent on these equidistributed meshes. Numerical results are presented to validate the theory and to demonstrate the efficiency of the proposed method.