An Exponentially Fitted Upwind Scheme for Singularly Perturbed Differential Equations With Mixed Shift Parameters

Abstract

This paper provides numerical solutions to a class of singularly perturbed differential–difference equations characterized by mixed shift parameters. The solutions of such problems exhibit sharp boundary layers near the endpoints of the spatial domain due to the presence of a small perturbation parameter ε(0 < ε ≪ 1). Consequently, classical numerical methods fail to give accurate results on uniform meshes. To address this challenge, we propose a numerical scheme that discretizes the problem using the Crank–Nicolson method in the temporal direction and an exponentially fitted finite difference scheme in the spatial direction, both on uniform meshes. Stability and convergence analyses confirmed that the proposed scheme is uniformly convergent with respect to the perturbation parameter ε, with second‐order accuracy in the temporal and spatial directions. Three model examples were presented for simulation, and the findings indicated that the theoretical analysis aligns with the practical results. Furthermore, the numerical results demonstrated that the proposed scheme outperforms several existing methods in the literature.