Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-diffusion Problems with a Discontinuous Initial Condition

This article presents the Richardson extrapolation techniques for solving singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions (DIC). The scheme uses- backward Euler for temporal derivatives on a uniform mesh and classical upwind finite difference method (FDM) for spatial derivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in both space and time variables. The method employs an upwind finite difference operator on a piecewise- uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. The numerical solution's accuracy is improved by using Richardson extrapolation techniques, which raises it from O(N-1 lnN +
t) to O(N-2 ln2 N +
t2) in the discrete maximum norm, where N is the number of spatial mesh intervals, and t is the size of the temporal step size. Parameter-uniform error estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.