Hybrid Difference Scheme for Singularly Perturbed Differential Equation with Discontinuous Source Term

Abstract

In this article, we investigate a hybrid difference scheme for finding the numerical solution of a singularly perturbed second-order reaction-diffusion problem with a discontinuous source term. Such types of problems arise in the modeling of semiconductor devices and geophysical fluid dynamics etc. Solutions of these types of problems are difficult to obtain due to the presence of boundary and interior layers. A hybrid difference scheme i.e., cubic spline method and central finite difference approach, are applied on a fine region and coarse region, respectively. Shishkin mesh is utilized to generate the mesh point for the given domain. We use a second-order hybrid difference operator at the point of discontinuity. The solution rapidly changes in the interior layers and boundary layer. Truncation error is studied, and the stability of the method is analyzed. The proposed method is implemented on two problems, and numerical results are compared with the existing method, which shows that the proposed method is efficient for reducing maximum absolute errors and increasing the rate of convergence.