Uniformly Convergent Second Order Numerical Method for a Class of Parameterized Singular Perturbation Problems

Abstract

In this article, a class of nonlinear singularly perturbed boundary value problems depending on a parameter are considered. To solve this class of problems; first we apply the backward Euler finite difference scheme on Shishkin type meshes [standard Shishkin mesh (S-mesh), Bakhvalov–Shishkin mesh (B–S-mesh)]. The convergence analysis is carried out and the method is shown to be convergent with respect to the small parameter and is of almost first order accurate on S-mesh and first order accurate on B–S-mesh. Then, to improve the accuracy of the computed solution from almost first order to almost second order on S-mesh and from first order to second order on B–S-mesh, the post-processing method namely, the Richardson extrapolation technique is applied. The proof for the uniform convergence of the proposed method is carried out on both the meshes. Numerical experiments indicate the high accuracy of the proposed method.