A numerical scheme to solve mixed parabolic-elliptic problem involving singular perturbation

Abstract

In this article, a singularly perturbed mixed-type problem of parabolic-elliptic is considered on a rectangular domain. The solution of the problem possesses both boundary and interior layers which appears in the spatial variable. To approximate the time derivative, first the implicit Euler scheme and then the Crank–Nicolson scheme on uniform mesh in time direction is used. For approximating the spatial derivative, we use the central difference scheme in the first sub-domain and a hybrid scheme in the second sub-domain on Shishkin type meshes (standard Shishkin mesh and Bakhvalov–Shishkin mesh). We prove that both the numerical schemes converge uniformly with respect to the perturbation parameter and are of second-order accurate. Thomas algorithm is used to solve the tri-diagonal system. The numerical results and the error bounds are illustrated through few tables and graphs.