Asymptotically Compatible Energy and Dissipation Law of the Nonuniform L2-1σ Scheme for Time Fractional Allen–Cahn Model

• Published in: Journal of scientific computing 2024-05, Vol.99 (2), p.46
• Main Authors: Liao, Hong-lin, Zhu, Xiaohan, Sun, Hong
• Format: Article
• Language: English
• Subjects: Algorithms; Asymptotic properties; Compatibility; Computational Mathematics and Numerical Analysis; Energy dissipation; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-024-02515-3

We build an asymptotically compatible energy of the variable-step L2- 1 σ scheme for the time-fractional Allen–Cahn model with the Caputo’s fractional derivative of order α ∈ ( 0 , 1 ) , under a weak step-ratio constraint τ k / τ k - 1 ≥ r ⋆ ( α ) for k ≥ 2 , where τ k is the k -th time-step size and r ⋆ ( α ) ∈ ( 0.3865 , 0.4037 ) for α ∈ ( 0 , 1 ) . It provides a positive answer to the open problem in Liao et al. (J Comput Phys 414:109473, 2020), and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen–Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2- 1 σ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy.