A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems

This article focuses on the numerical solution of a time-dependent parabolic problem that exhibits singular perturbations and involves two perturbation parameters. To address this problem, a fitted mesh finite difference method is developed. In numerical discretization, the implicit Crank-Nicolson technique is employed to discretize the time derivative using a uniform mesh. As for the spatial derivative, a hybrid finite difference scheme known as the adaptive fitted mesh of the Shishkin type is utilized. The study also includes a discussion on a priori bounds for the continuous solution and its derivatives. The proposed method is proven to be uniformly convergent of order two in both time and space. Theoretical analysis and simulations on various test examples confirm the scheme’s accuracy and convergence properties.