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, which is effectively managed by adopting a discretization approach for the time-fractional derivative, where Alikhanov's high-order L2-1$_\sigma$ formula is applied on a non-uniform fitted mesh, resulting in successful tackling of the singularity. A high-order two-dimensional compact operator is implemented to approximate the spatial variables. The alternating direction implicit (ADI) approach is then employed 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, has been conducted comprehensively, and it has shown that method is convergent with an order $\mathcal O\left(N_t^{-\min\{3-\alpha,\theta\alpha,1+2\alpha,2+\alpha\}}+h_x^4+h_y^4\right)$, 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 spatial mesh widths. Moreover, the parameter $\theta$ is utilized in the construction of the fitted mesh.

翻译：本研究提出了一种求解二维时间分数阶对流扩散(TFCD)方程的新型高阶数值方法。采用Caputo定义刻画时间分数阶导数。所考虑问题在初始时刻($t=0$)存在弱奇异性，通过采用时间分数阶导数的离散化方法有效处理该奇异性：在非均匀拟合网格上应用Alikhanov高阶L2-1$_\sigma$公式，成功克服了奇异性问题。采用高阶二维紧致算子逼近空间变量，进而利用交替方向隐式(ADI)方法将二维问题分解为两个独立的一维问题进行求解。全面开展了包含稳定性与收敛性的理论分析，证明该方法具有$\mathcal O\left(N_t^{-\min\{3-\alpha,\theta\alpha,1+2\alpha,2+\alpha\}}+h_x^4+h_y^4\right)$阶收敛性，其中$\alpha\in(0,1)$表示分数阶导数阶数，$N_t$为时间离散参数，$h_x$和$h_y$分别表示空间网格宽度，参数$\theta$用于构造拟合网格。