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This study presents a novel high-order numerical method designed for solvingthe two-dimensional time-fractional convection-diffusion (TFCD) equation. TheCaputo definition is employed to characterize the time-fractional derivative. Aweak singularity at the initial time ($t=0$) is encountered in the consideredproblem, which is effectively managed by adopting a discretization approach forthe time-fractional derivative, where Alikhanov's high-order L2-1$_\sigma$formula is applied on a non-uniform fitted mesh, resulting in successfultackling of the singularity. A high-order two-dimensional compact operator isimplemented to approximate the spatial variables. The alternating directionimplicit (ADI) approach is then employed to solve the resulting system ofequations by decomposing the two-dimensional problem into two separateone-dimensional problems. The theoretical analysis, encompassing both stabilityand convergence aspects, has been conducted comprehensively, and it has shownthat method is convergent with an order $\mathcalO\left(N_t^{-\min\{3-\alpha,\theta\alpha,1+2\alpha,2+\alpha\}}+h_x^4+h_y^4\right)$,where $\alpha\in(0,1)$ represents the order of the fractional derivative, $N_t$is the temporal discretization parameter and $h_x$ and $h_y$ represent spatialmesh widths. Moreover, the parameter $\theta$ is utilized in the constructionof the fitted mesh.

Error analysis of a high-order fully discrete method for two-dimensional  time-fractional convection-diffusion equations exhibiting weak initial  singularity