A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

The conservative Allen-Cahn equation satisfies three important physical properties, namely the mass conservation law, the energy dissipation law, and the maximum bound principle. However, very few numerical methods can preserve them at the same time. In this paper, we present a multi-physical structure-preserving method for the conservative Allen-Cahn equation with nonlocal constraint by combining the averaged vector field method in time and the central finite difference scheme in space, which can conserve all three properties simultaneously at the fully discrete level. We propose an efficient linear iteration algorithm to solve the presented nonlinear scheme and prove that the iteration satisfies the maximum bound principle and a contraction mapping property in the discrete L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{L}<^>{\varvec{\infty }}$$\end{document} norm. Furthermore, concise error estimates in the maximum norm are established on non-uniform time meshes. The theoretical findings of the proposed scheme are verified by several benchmark examples, where an adaptive time-stepping strategy is employed.