a fourth-order optimal numerical approximation and its convergence for singularly perturbed time delayed parabolic problems

This paper presents a numerical solution for a time delay parabolic problem (reactiondiffusion) containing a small parameter. the numerical method combines the implicit crank–nicolson scheme for the time  derivative on the uniform mesh and the central difference scheme for the spatial derivative on the shishkin type meshes. it is shown to be secondorder uniformly convergent in time and space. then richardson extrapolation technique is applied to enhance the accuracy from secondorder to fourthorder. the error analysis is carried out, and the method is proved to be uniformly convergent. these two methods are applied to two test examples in support of the theoretical results.