A fitted mesh robust numerical method and analysis for the singularly perturbed parabolic PDEs with a degenerate coefficient

In this study, we present a nearly second-order central finite difference approach for solving a singularly perturbed parabolic problem with a degenerate coefficient. The approach uses a Crank–Nicolson method to discretize the time direction on the uniform mesh and a second-order central finite difference method on the Shishkin mesh in the space direction. The solution to the problem shows a parabolic boundary layer around $x=0$. Our error estimates indicate that the suggested approach is nearly second-order $\varepsilon$-uniformly convergent both in space and time directions. Some numerical results have been generated to validate the theoretical findings. Extensive comparisons have been carried out, demonstrating that the current approach is more accurate than previous methods in the literature.