An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

Abstract

In this paper we consider a 1D parabolic singularly perturbed reaction-convection–diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter ε2) and the convection term (multiplied by the parameter μ) in the differential equation (ε∈(0,1], μ∈[0,1], μ≤ε). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in ε and μ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed.