Linearly Implicit Schemes Preserve the Maximum Bound Principle and Energy Dissipation for the Time-fractional Allen–Cahn Equation

Abstract

This paper presents two highly efficient numerical schemes for the time-fractional Allen–Cahn equation that preserve the maximum bound principle and energy dissipation in discrete settings. To this end, we utilize a generalized auxiliary variable approach proposed in a recent paper (Ju et al. in SIAM J Numer Anal 60:1905–1931, 2022) to reformulate the governing equation into an equivalent system that follows a modified energy functional and the maximum bound principle at each continuous level. By combining the L1-type formula of the Riemann–Liouville fractional derivative with the Crank–Nicolson method, we construct two novel linearly implicit schemes by introducing the first- and second-order stabilized terms, respectively. These schemes are proved to be energy stable and maximum bound principle preserving on nonuniform time meshes with the help of the discrete orthogonal convolution technique. In addition, we obtain the unique solvability of the proposed schemes without any time-space step ratio. Finally, we report extensive numerical results to verify the correctness of the theoretical analysis and the performance of the proposed schemes in long-time simulations.