Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations

In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time-dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of O(Delta t + N-2 (In N)(2)), where Delta t and N denote the time step and number of mesh-intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of O(Delta t(2) + N-2 (ln N)(2)). Some numerical tests are performed to illustrate the high-order accuracy and parameter uniform convergence obtained with the proposed numerical methods.