A parameter numerical scheme for singularly perturbed parabolic differential-difference problems with large time lag

This work presents a robust numerical algorithm for singularly perturbed parabolic differential-difference problems with large time delay. The scheme begins with the approximation of retarded terms in the spatial direction via Taylor’s series expansion. The resulting problem is discretized using the Crank-Nicolson scheme for the temporal derivative and a hybrid method that combines the non-standard upwind finite difference with a fitted mesh approach for the spatial derivative. The uniform convergence and stability characteristics of the scheme are thoroughly investigated. It is shown that the method achieves parameter-uniform convergence with second-order accuracy in time and first-order accuracy in space. Numerical test examples are conducted, and the results demonstrate the effectiveness and improved accuracy of the proposed scheme compared to existing methods in the literature.