In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.

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A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. In Miller et al. (J Appl Numer Math 35(4):323-337, 2000) the authors discussed problems that arises naturally in the context of models of simple semiconductor devices. Due to the discontinuity, interior layers appear in the solution. The problem is solved using a hybrid difference scheme on a Shishkin mesh. We prove that the method is second order convergent in the maximum norm, independently of the diffusion parameter. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

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