Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

The purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.

The small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.

The proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.

The use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.

To validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.