Efficient numerical schemes on modified graded mesh for singularly perturbed parabolic convection-diffusion problems

In this study, numerical approaches to the singularly perturbed problems of convection diffusion type are presented. The backward Euler method is applied to a uniform mesh in the temporal domain, while in the spatial domain, we utilize both the hybrid midpoint finite difference scheme and the high order via differential identity expansion scheme on a modified graded mesh. The solution to the problem introduces a boundary layer on the right side of the domain. Both of the above methods are proven to have identical convergence with respect to the perturbation parameter. We also provide numerical results in order to verify the theoretical conclusions. We demonstrate that the applied approaches provide uniform convergence of first-order in the temporal variable and second-order up to a logarithmic factor with respect to the spatial variable.

کلیدواژه ها:

Perturbation problems ، Uniform convergence ، Modified graded mesh ، Boundary layers ، Hybrid midpoint finite difference scheme ، HODIE finite difference scheme