A robust uniformly convergent finite difference scheme for the time-fractional singularly perturbed convection-diffusion problem

Abstract

The design and analysis of a finite difference method for solving a class of time-fractional singularly perturbed convection-diffusion problems is the subject of this paper. The time-fractional derivative is taken in the Caputo sense with the order α∈(0,1). The solution to this class of problems possesses weak singularity in the vicinity of the initial time t=0 and an exponential boundary layer at the right lateral surface as the singular perturbation parameter ε→0. To tackle the initial layer at time t=0, a classical L1 finite difference scheme is employed on a graded mesh to discretize the time-fractional derivative. Further, a standard upwinding procedure in the spatial direction is used on a piecewise uniform Shishkin mesh. It is shown that the resultant fully discrete scheme is parameter uniformly convergent with order of convergence O(M−p+N−1log⁡N), where p=min⁡{2−α,rα}, r is the graded mesh exponent and N,M are the number of mesh subintervals in space and time direction respectively. Lastly, numerical results illustrate that the proposed scheme is simple, efficient, robust and accurate as well as validate the error estimates presented in the paper.