Efficient Structure-Preserving Scheme for the Space Fractional Allen-Cahn Equation with Logarithmic Flory-Huggins Potential

In this paper, a numerical method is proposed for solving the space fractional Allen-Cahn equation with logarithmic Flory-Huggins potential. Due to the strong nonlinearity of potential function and singularity of logarithmic term, how to constructed an effective scheme that preserves both the unconditional discrete maximum principle and unconditional original energy stability is a great challenge for this problem. To overcome these difficulties, the stabilized method is applied to construct a structure-preserving scheme, where the weighted and shifted Gr & uuml;nwald difference formula is used to approximate the Riesz fractional derivative. The advantages of this method are that the nonlinear logarithmic term is explicitly treated, the proposed scheme is linear and easy to implement and it preserves the unconditional original energy stability. Then, for any time step, the unique solvability of scheme, discrete maximum principle preserving and original energy stability are all rigorously proved. Moreover, the detailed error estimate in maximum norm is given. In addition, adaptive time-stepping algorithm is used to improve computational efficiency in long time simulations. In numerical experiments, the effects of fractional order alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} on the phase change process are also studied. The numerical results verify the effectiveness of the proposed algorithms and the correctness of the theoretical analysis.