Cahn-Hilliard方程的自适应间断有限体积元法

数学物理学报 ›› 2023, Vol. 43 ›› Issue (4): 1255-1268.

Cahn-Hilliard方程的自适应间断有限体积元法

曾纪尧,李剑^*()

陕西科技大学数学与数据科学学院西安 710021

收稿日期:2022-04-06 修回日期:2022-09-13 出版日期:2023-08-26 发布日期:2023-07-03
通讯作者: 李剑 E-mail:jianli@sust.edu.cn
基金资助:
国家自然科学基金(11771259);陕西省人工智能联合实验室(2022JC-SYS-05);陕西省教育厅创新团队项目(21JP013);部分资助和陕西省自然科学基础研究计划重点项目(2023-JC-ZD-02)

An Adaptive Discontinuous Finite Volume Element Method for the Cahn-Hilliard Equation

Zeng Jiyao,Li Jian^*()

School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021

Received:2022-04-06 Revised:2022-09-13 Online:2023-08-26 Published:2023-07-03
Contact: Jian Li E-mail:jianli@sust.edu.cn
Supported by:
NSF of China(11771259);Shaanxi Provincial Joint Laboratory of Artificial Intelligence(2022JC-SYS-05);Innovative Team Project of Shaanxi Provincial Department of Education(21JP013);Shaanxi Province Natural Science Basic Research Program Key Project(2023-JC-ZD-02)

摘要/Abstract

摘要：

Cahn-Hilliard 方程是一类重要的四阶非线性扩散方程, 具有丰富的物理背景和深刻的研究价值, 但方程的非线性势项 $ f(u) $ 的存在以及小参数 $ \epsilon $ 导致的强刚性会给数值模拟带来诸多挑战, 因此设计高效、准确的数值方案满足方程离散能量定律是非常重要的. 间断有限体积元方法 (DFVEM) 采用低阶元, 具有精度高、操作简单、适合工程应用的网格自适应等优点. 该文对Cahn-Hilliard 方程利用 DFVEM 结合全隐格式进行求解, 证明了全离散格式质量守恒和能量耗散的重要理论结果. 数值实验提出一种自适应时间步进策略, 验证了方法的有效性.

关键词: Cahn-Hilliard 方程, 间断有限体积元法, 离散能量耗散

Abstract:

The Cahn-Hilliard equation is an important class of fourth-order nonlinear diffusion equations with rich physical background and profound research value. In the numerical simulation, the existence of the nonlinear potential term $ f(u) $ of the equation and the strong rigidity caused by small parameter $ \epsilon $ will bring many challenges, so it is important to design efficient and accurate numerical schemes to satisfy the discrete energy law of the equations. In this paper, the Cahn-Hilliard equation is solved by the discontinuous finite volume element method (DFVEM) combined with the fully implicit scheme, and the important theoretical results of mass conservation and energy dissipation in the fully discrete scheme are proved. At the same time, the semi-discrete format error estimates are given. Finally, numerical experiments propose an adaptive time stepping strategy and verify the effectiveness of the method.

Key words: The Cahn-Hilliard equation, Discontinuous finite volume element method, Discrete energy dissipation

中图分类号:

O241.82

曾纪尧,李剑. Cahn-Hilliard方程的自适应间断有限体积元法[J]. 数学物理学报, 2023, 43(4): 1255-1268.

Zeng Jiyao,Li Jian. An Adaptive Discontinuous Finite Volume Element Method for the Cahn-Hilliard Equation[J]. Acta mathematica scientia,Series A, 2023, 43(4): 1255-1268.

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