In this paper, a two-dimensional time-fractional Allen-Cahn equation with the Caputo derivative of order \(\alpha \in (0, 1)\) is considered. In order to facilitate the construction of efficient numerical algorithms, we first transform the original equation into an equivalent integral equation. Because the solution of this equation has weak singularity at the initial time, one of the most effective ways to deal with it is to use numerical integration formulas with non-uniform meshes to approximate the fractional integral operator contained in it. Therefore, we have constructed a new non-uniform mesh that is superior to the strategies in existing literatures, and combined the fractional rectangle method and the fractional trapezoid method to obtain two new numerical integration formulas for approximating the fractional integral. Next, by applying the central difference formula to the spatial derivative, two efficient numerical schemes for solving the equivalent integral equation can be obtained. Furthermore, we conduct a detailed study of the discrete maximum principle and energy decay under certain conditions for the fractional trapezoidal numerical scheme. Meanwhile, the stability and convergence have also been rigorously proven, and the results show that its time and spatial convergence orders are both second-order. Finally, we provide a numerical example that demonstrates the superiority of our constructed non-uniform mesh over existing meshes.
