Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

• Published in: Advances in computational mathematics Vol. 46; no. 2
• Main Authors: Ji, Bingquan, Liao, Hong-lin, Zhang, Luming
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Computer simulation; Finite element method; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Maximum principle; Visualization; 74A50; 65M06; Time-fractional Allen-Cahn equation; 35Q99; Discrete maximum principle; Fast L1 formula; Adaptive time-stepping strategy; Sharp error estimate; 65M12
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09782-2

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo’s derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time steps are always incorporated into our approaches so that a adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.