High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay

In this article we study numerical approximation for singularly perturbed parabolic partial differential equations with time delay. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. The problem is discretized by a hybrid scheme on a generalized Shishkin mesh in spatial direction and the implicit Euler scheme on a uniform mesh in time direction. We then design a Richardson extrapolation scheme to increase the order of convergence in time direction. The resulting scheme is proved to be second order accurate in time direction and fourth order (with a factor of logarithmic type) accurate in spatial direction. Numerical experiments are performed to support the theoretical results.