Two types of asymptotically compatible energies are constructed for the variable-step L1 scheme for the time-fractional Allen-Cahn model with Caputo’s fractional derivative of order \(\alpha \in (0,1)\) . An energy of the time-fractional Allen-Cahn model is called asymptotic compatibility if it approaches that of the classical Allen-Cahn model when the fractional order \(\alpha \rightarrow 1^-\) . By splitting the L1 formula into a local part and a nonlocal part, we construct two discrete gradient structures by exploring the logarithmic convexity and algebraic convexity of associated kernels, respectively. The nonlinear implicit L1 and linearly implicit L1-SAV schemes are then investigated for the time-fractional Allen-Cahn model, and new discrete energy dissipation laws are established under some mild step-ratio constraints. Extensive numerical tests are provided to examine the accuracy, energy behaviors, and solution behaviors of our numerical solvers in the long-time simulations. They suggest that the asymptotically compatible energy constructed from the algebraic convexity approximates the original energy faster than that constructed from the logarithmic convexity. It seems that both methods monotonically converge to the correct steady-state solution for any initial data.
