Stability and Error Analysis of an Efficient Numerical Method for Convection Dominated Parabolic PDEs with Jump Discontinuity in Source Function on Modified Layer-Adapted Mesh

Abstract

This paper introduces a novel approach for analyzing efficient numerical solution of a class of singularly perturbed parabolic convection-diffusion PDEs having a finite jump discontinuity mostly in the source function at the interior of the spatial domain. These PDEs often appear in mathematical modeling of the semiconductor devices; and solutions of such problems usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. The prime objectives of this paper are to overcome the theoretical challenge related to the monotonocity of the finite difference operator which utilizes one-sided second-order difference operators at the interface point; and also to establish higher-order numerical approximation in space, regardless of smaller and larger values of the parameter ε. Proving discrete maximum principle of the proposed difference operator is found to be challenging task on the standard Shishkin mesh adapted to both boundary and weak interior layers. We overcome this difficulty by constructing a special non-uniform mesh, called modified layer-adapted mesh; and hereby establish the stability as well as the parameter-uniform error estimate in the discrete supremum norm. Finally, the theoretical estimate is verified with numerical results for test examples with and without the exact solution. Moreover, numerical results are obtained for the semilinear PDEs and also compared with the implicit upwind scheme to exhibit the efficiency of the proposed algorithm.