richardson extrapolation technique on a modified graded mesh for singularly perturbed parabolic convection-diffusion problems

In this paper, we focus on investigating a post-processing technique designed for one-dimensional singularly perturbed parabolic convection-diffusion problems that demonstrate a regular boundary layer. we use a back-ward euler numerical approach for time derivatives with uniform mesh in the temporal direction, and a simple upwind scheme is used for spatial derivatives with modified graded mesh in the spatial direction. in this study, we demonstrate the effectiveness of the richardson extrapolation technique in enhancing the ε-uniform accuracy of simple upwinding within the discrete supremum norm, as evidenced by an improvement from o(n^−1 ln(1/ε) + △θ) to o(n^−2 ln² (1/ε) + △θ²). furthermore, to validate the theoretical findings, computational experiments are conducted for two test examples by applying the proposed technique.