Discrete Gradient Methods Preserve the Energy Dissipation Law and Maximum Bound Principle of the Time-Fractional Allen–Cahn equation with General Nonlinear Potential

In this paper, we design two classes of numerical schemes for the time-fractional Allen–Cahn equation based on the discrete gradient method and the L1-type formula of the Riemann–Liouville fractional derivative. The suggested schemes are as accurate and efficient as a recent work [H. Liao, T. Tang and T. Zhou, An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen–Cahn equation, SIAM J. Sci. Comput. 43 2021, 5, A3503–A3526] for the time-fractional Allen–Cahn equation with a double-well potential function, but can address the time-fractional Allen–Cahn equation with a general potential function. Thanks to the orthogonal convolution technique on nonuniform time mesh: (i) the first type of schemes unconditionally preserves the discrete energy dissipation law; (ii) the second type of schemes is constructed by introducing the fixed-point iteration and stabilized factor method for the first scheme, and proven to be maximum bound principle preserving. Finally, extensive numerical comparisons are reported to verify the efficiency of the proposed schemes and the correctness of the theoretical results.