Error analysis of a high-order compact ADI scheme for 2D time-fractional convection-diffusion equation with weakly singular solutions

In this paper, we propose a high-order numerical method for the 2D time-fractional convection-diffusion equation with weakly singular solutions, where the time Caputo fractional derivative is approximated by using the $L2$- $1_{\sigma }$ formula on a nonuniform graded mesh, and the space derivatives are discretized by a fourth-order compact finite difference scheme on a uniform mesh. The fully discrete compact ADI scheme is established by adding a high-order term. The stability and the convergence analyses of the scheme are analyzed in $L^2$-norm by using the discrete energy method. It has been proved that the introduced numerical scheme has spatial fourth-order convergence, and temporal optimal $(1+\alpha )$- $th$ order convergence. The numerical results show that the error estimates are sharp.