Numerics of singularly perturbed differential equations

Abstract

The main purpose of this report is to carry out the effect of the various numerical methods for solving singular perturbation problems on non-uniform meshes. When a small parameter epsilon known as the singular perturbation parameter is multiplied with the higher order terms of the differential equation, then the differential equation becomes singularly perturbed. In this type of problems, there are regions where the solution varies very rapidly known as boundary layers and the region where the solution varies uniformly known as the outer region. Standard finite difference/element methods are applied on the singularly perturbed differential equation on uniform mesh give unsatisfactory result as epsilon tends to zero. Due to presence of boundary layer, standard difference schemes unable to capture the layer behaviour until the mesh parameter and perturbation parameter are of the same size which results vast computational cost. In order to overcome this difficulty, we adapt non-uniform meshes. The Shishkin mesh and the adaptive mesh are two widely used special type of non-uniform meshes for solving singularly perturbed problem. Here, in this report singularly perturbed problems namely convection-diffusion and reaction-diffusion problems are considered and solved by various numerical techniques. The numerical solution of the problems are compared with the exact solution and the results are shown in the shape of tables and graphs to validate the theoretical bounds.