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In this paper, the parabolic singularly perturbed convection-diffusion problem is discretized by using the backward Euler method in time and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh in space. This method is shown to be first-order convergent in time and space. Finally numerical experiments confirm the theoretical results.

The Midpoint Upwind Scheme on the Bakhvalov-Shishkin Mesh for Parabolic Singularly Perturbed Problems