An efficient uniformly convergent numerical scheme for singularly perturbed semilinear parabolic problems with large delay in time

This paper introduces a parameter uniform numerical approach for semilinear singularly perturbed partial differential equations with time delay. The semilinear property is treated with the help of the quasilinearization technique. For the temporal direction, the Crank-Nicholson scheme is used on the uniform mesh and to handle the spatial derivative term, the upwind scheme on both Shishkin mesh and Bakhvalov-Shishkin mesh is used. To validate the theoretical findings and to show the efficacy of the proposed scheme, two different kinds of examples are provided. The corresponding maximum absolute errors and rates of convergence are tabulated. The proposed method is proved to be first-order uniformly convergent, independent of the perturbation parameter and robust.