Maximum Principle Preserving Schemes for Binary Systems with Long-Range Interactions

We study some maximum principle preserving and energy stable schemes for the Allen–Cahn–Ohta–Kawasaki model with fixed volume constraint. With the inclusion of a nonlinear term f(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\phi )$$\end{document} in the Ohta–Kawasaki free energy functional, we show that the Allen–Cahn–Ohta–Kawasaki dynamics is maximum principle preserving. We further design some first order energy stable numerical schemes which inherit the maximum principle preservation in both semi-discrete and fully-discrete levels. Furthermore, we apply the maximum principle preserving schemes to a general framework for binary systems with long-range interactions. We also present some numerical results to support our theoretical findings.