A Robust Numerical Method for a Singularly Perturbed Parabolic Convection-Diffusion Problem with a Degenerating Convective Term and a Discontinuous Right-Hand Side

Abstract

In this paper we consider the efficient numerical approximation of a singularly perturbed parabolic convection-diffusion problem having a convective term which degenerates inside the domain, in the case that the right-hand side of the differential equation is discontinuous on the degeneration line. For small values of the diffusion parameter \({\varepsilon }^{2}\) (\(\varepsilon \in (0,1]\)), in general, the exact solution has an interior layer in a neighborhood of the degeneration line. We construct a classical finite difference scheme combining the implicit Euler method in time, defined on a uniform mesh, and the first order upwind scheme in space, defined on a piecewise-uniform grid condensing in a neighborhood of the interior layer. Then, the method is an \(\varepsilon \)-uniformly convergent scheme of first order in time and almost first order in space. We show the numerical results for a test problem, confirming in practice the theoretical results.