Fitted computational method for convection dominated diffusion equations with shift arguments

Abstract

A time dependent singularly perturbed convection diffusion equation involving shift parameters are considered. The terms containing the shifts are treated using Taylor series approximation up to second order terms. Classical numerical methods developed for solving regular problems fail to give good approximate solution and become unstable while applied for the considered problem. In this paper, numerical scheme is developed using ?-method for semi-discretizing in time derivative; in spatial discrtization fitted operator finite difference method is applied by inducing exponential fitting parameter. To accelerate the convergence of the scheme, Richardson extrapolation technique is applied in spatial discretization. Existence of unique discrete solution is guaranteed by establishing the discrete comparison principle. The proposed scheme is stable for all values of the perturbation parameter. A uniformly convergent solution is obtained by Richardson extrapolation method which provides second order accuracy under some condition. Test examples are considered for validating the theoretical results numerically.