Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients

Abstract

This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are presented to validate the theoretical results.