The Exponential SAV Approach for the Time-Fractional Allen–Cahn and Cahn–Hilliard Phase-Field Models

In this paper, we take a consideration of a class of time-fractional phase-field models including the Allen–Cahn and Cahn–Hilliard equations. Based on the exponential scalar auxiliary variable (ESAV) approach, we construct two explicit time-stepping schemes, in which the fractional derivative is discretized by L1 and L1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L1+$$\end{document} formulas respectively. It is worth to mentioning that our novel schemes are effective for the completely decoupled computations of the phase variable ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and the auxiliary variable R. In fact, the above two schemes admit energy dissipation law on general nonuniform meshes, which is inherent property in the continuous level. Finally, several numerical experiments are carried out to verify the accuracy and efficiency of our proposed methods.