Arbitrarily High Order and Fully Discrete Extrapolated RK–SAV/DG Schemes for Phase-field Gradient Flows

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作者
Tao Tang
Xu Wu
Jiang Yang
机构
[1] BNU-HKBU United International College,Division of Science and Technology
[2] Southern University of Science and Technology,SUSTech International Center for Mathematics
[3] Harbin Institute of Technology,School of Mathematics
[4] Southern University of Science and Technology,Department of Mathematics
[5] Southern University of Science and Technology,Guangdong Provincial Key Laboratory of Computational Science and Material Design
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Phase-field models; Gradient flows; Energy stability; Convergence and error analysis; Allen–Cahn equation; Cahn–Hilliard equation;
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摘要
In this paper, we construct and analyze a fully discrete method for phase-field gradient flows, which uses extrapolated Runge–Kutta with scalar auxiliary variable (RK–SAV) method in time and discontinuous Galerkin (DG) method in space. We propose a novel technique to decouple the system, after which only several elliptic scalar problems with constant coefficients need to be solved independently. Discrete energy decay property of the method is proved for gradient flows. The scheme can be of arbitrarily high order both in time and space, which is demonstrated rigorously for the Allen–Cahn equation and the Cahn–Hilliard equation. More precisely, optimal L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-error bound in space and qth-order convergence rate in time are obtained for q-stage extrapolated RK–SAV/DG method. Several numerical experiments are carried out to verify the theoretical results.
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