Energy dissipation and maximum-bound principle of the variable-step L2-1σ scheme for the time-fractional Allen–Cahn equation with general nonlinear potential

Abstract

In this study, we focus on a numerical scheme that maintains both the energy-dissipation law and the maximum-bound principle for the time-fractional Allen–Cahn equation with a general nonlinear potential. We propose a stabilized linear iterative method, using the variable-step L2-$1_{\sigma }$ formula for the discretization of the Caputo fractional derivative in time and the central finite difference method for the spatial Laplacian. Furthermore, graded meshes are utilized to address the initial singularity and adaptive strategies are used to capture multiscale behavior. The proposed method is demonstrated to preserve the energy-dissipation law and maximum-bound principle in discrete settings. With the help of the maximum boundedness of the numerical solution, we derive the $L^{\infty }$-norm error estimate of the proposed scheme by using the discrete fractional Gönwall inequality. Finally, we provide extensive numerical results to verify the theoretical results and computational efficiency of the proposed scheme.