Unconditionally Energy-Dissipation- and Maximum-Bound-Principle-Preserving Scheme for the Time-Fractional Allen-Cahn Equation

Abstract

In this paper, we propose and analyze a class of linearly implicit energy stable scheme for the nonlinear time-fractional Allen-Cahn equation with a general potential function, where the temporal and spatial derivatives are approximated by the variable-step L1 method and the central difference method, respectively. The proposed scheme is proved to be unconditionally energy-dissipation-preserving and maximum-bound- principle-preserving in discrete settings with the help of the discrete orthogonal convolution technique. Thanks to the maximum-normal boundness of numerical solution and the discrete fractional Grönwall inequality, we obtain the convergence in $L^2$ norm of the proposed scheme. In practical computation, we utilize the graded mesh and the adaptive mesh to simulate the time-fractional Allen-Cahn equation to capture the multi-scale behaviors and enhance computational efficiency. Extensive numerical comparisons are provided to verify the correctness and efficiency of the proposed scheme in long-time computations.