A High-Order Discrete Energy Decay and Maximum-Principle Preserving Scheme for Time Fractional Allen–Cahn Equation

• Published in: Journal of scientific computing Vol. 96; no. 2; p. 39
• Main Authors: Zhang, Guoyu, Huang, Chengming, Alikhanov, Anatoly A., Yin, Baoli
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2023; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Computational Mathematics and Numerical Analysis; Decay; Energy dissipation; Finite difference method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Numerical analysis; Principles; Review; Theoretical; Difference scheme; Discrete energy decay; Time-fractional Allen–Cahn equation; Shifted fractional trapezoidal rule; Maximum-principle
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-023-02263-w

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen–Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1 σ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results.