A second-order finite difference scheme for singularly perturbed initial value problem on layer-adapted meshes

Abstract

In this paper, a second-order singularly perturbed initial value problem is considered. A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes. The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm. It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter. To show the performance of the proposed method, it is applied on few test examples which are in agreement with the theoretical results. Furthermore, existing results are also compared to show the robustness of the proposed scheme.