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

This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time (\(t=0\)) is encountered in the considered problem. To overcome this, we consider the high-order L2-1\(_\sigma \) formula on a suitably designed non-uniform fitted mesh, to discretize the time-fractional derivative. Further, a high-order two-dimensional compact operator is developed to approximate the spatial variables. Moreover, an alternating direction implicit (ADI) approach is designed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, is conducted comprehensively. More precisely, it is shown that method is convergent of order \(\mathcal O\left( {N_t^{-\min \{3-\alpha ,\theta \alpha ,1+\alpha ,2\}}}+h_x^4+h_y^4\right) \), where \(\alpha \in (0,1)\) represents the order of the fractional derivative, \(\theta \) is a parameter which is utilized in the construction of the fitted mesh, \(N_t\) is the temporal discretization parameter, and \(h_x\) and \(h_y\) represent the spatial mesh widths. The numerical outcomes for three test problems, each featuring the nonsmooth solution, verified the theoretical findings. Further, the proposed method on fitted meshes exhibits superior numerical accuracy in comparison to the existing methods.