A Midpoint Upwind Numerical Scheme for Singularly Perturbed Differential Difference Equations

Abstract

In this work, we consider a class of singularly perturbed differential‐difference equations with small shift parameters in the convection and reaction terms, which frequently arise in applied mathematics and engineering. The presence of a small diffusion parameter ε , (0 < ε ≪ 1) causes the solution of the considered problem to exhibit steep gradients (boundary layers), making classical numerical methods ineffective. To address this challenge, we formulate and analyze a fitted numerical scheme, employing the implicit Euler method in time and a midpoint upwind finite difference approach in space on uniform meshes. Stability and convergence analyses confirm second‐order accuracy in both directions after Richardson extrapolation. Numerical experiments validate the theoretical results and demonstrate that the proposed method outperforms several existing methods in the literature.