A high-order numerical method is proposed for solving a two-dimensional time-fractional diffusion equation with weak singularity at the initial time. The method employs L2-1 \(_\sigma \) scheme on a graded temporal mesh for discretization of time fractional derivative and a compact alternating direction implicit (ADI) scheme for discretization of space derivatives. The unconditional stability and convergence of the method in the \(H^1\) -norm are established theoretically. Numerical experiments are carried out and confirm a fourth-order spatial accuracy and temporal convergence rate of \(\min \{r\alpha , 1+\alpha \}\) , where \(\alpha \in (0,1)\) denotes the order of the fractional derivative. A comparison is made between the present ADI method and the Non-ADI scheme to justify the advantage of our method. It is shown that the present ADI method attains higher accuracy and requires less computational time. The graded mesh approach effectively handles the weak singularity at the initial time \(t = 0\) .
