An efficient numerical scheme and its analysis for the multiterm time-fractional convection-diffusion-reaction equation

This article aims at developing a robust numerical method based on graded mesh for solving the multiterm time-fractional convection-diffusion-reaction (TFCDR) equation whose solution very likely exhibits a weak singularity at the initial time. The time-fractional derivative in the model problem is described in terms of Liouville-Caputo. In order to handle the weak singularity at the initial time, we use a graded mesh technique for discretization of multiterm temporal fractional derivatives. The space derivatives are approximated by means of a compact finite difference (CFD) method. The proposed method is analyzed for its stability and convergence. Two numerical examples are considered to demonstrate the applicability and accuracy of the method. It is shown that the proposed graded mesh technique provides an optimal rate of convergence in time direction for the problem with nonsmooth exact solution, while the method on the uniform mesh yields a nonoptimal rate of convergence.