The Midpoint Upwind Finite Difference Scheme for Time-Dependent Singularly Perturbed Convection-Diffusion Equations on Non-Uniform Mesh

A numerical approach is proposed to examine the singularly perturbed time dependent convection-diffusion equation in one space dimension on a rectangular domain. The solution of considered problem exhibits a boundary layer on the right side of the domain. We semidiscretize the continuous problem by means of backward Euler finite difference method in the temporal direction. The semi-discretization process yields a set of ordinary differential equations at each time level. A resulting set of ordinary differential equations are discretized by using midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost of second order accurate in the coarse mesh and almost of first order accurate in fine mesh in the spatial direction. First order accuracy is achieved in the temporal direction. An extensive amount of analysis has been carried out in order to obtain uniform convergence of the method. Finally, we have found that the method is uniformly convergent with respect to the singular perturbation parameter i.e. e-uniform. Some numerical experiments have been carried out to validate the predicted theory.