Robust numerical methods for two-parameter singularly perturbed parabolic problems with non-smooth data

A uniformly convergent numerical scheme is developed to address two-parameter singularly perturbed parabolic equations involving discontinuous convection coefficients and source terms. These problems are challenging because the small perturbation parameters cause sharp boundary and interior layers, while discontinuities further complicate the solution structure, often leading to numerical instability and loss of accuracy in conventional methods. To overcome these challenges, the present study first discretizes the time derivative using the implicit Euler method, ensuring temporal stability. For the spatial variable, a non-standard finite difference approach is designed to achieve uniform convergence with respect to the perturbation parameters. Furthermore, Richardson extrapolation is employed to enhance both the convergence rate and the accuracy of the proposed method. The theoretical analysis confirms the scheme’s uniform convergence, while numerical experiments validate its robustness and precision even in the presence of strong layer behavior and discontinuous coefficients. The results reveal that the proposed technique not only produces stable and accurate solutions across all perturbation ranges but also provides a reliable computational framework for solving complex singularly perturbed parabolic models encountered in physical, chemical, and engineering processes. These findings contribute to advancing efficient numerical modeling strategies for problems where discontinuities and multiple small parameters coexist, aligning with the growing demand for reliable simulation tools in applied sciences.