Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation
Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation
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Cahn-Hilliard 方程隐式-显式方法的稳定性和收敛性分析
DOI：
10.1090/mcom/3704
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发表时间：
2021-09
影响因子：
2
通讯作者：
Tao Tang
中科院分区：
文献类型：
--
作者：
Dong Li;Chaoyu Quan;Tao Tang
隐-显方法已成功地用于相场问题的有效数值模拟，如Cahn-Hilliard方程或薄膜型方程。由于小耗散系数的影响，导致了极大值原理和刚度的缺乏，现有的理论分析大多依赖于增加额外的稳定项，缓和非线性或引入辅助变量，这隐含地改变了问题的结构或以微妙的方式以精度换取稳定性。在这项工作中，我们引入了一个强大的理论框架，直接分析的稳定性和准确性的标准隐式显式方法没有稳定或任何其他修改。以Cahn-Hilliard方程为例，在一定的时间步长约束下，对原半离散格式进行了严格的稳定性和收敛性分析。这些解决了几个问题，一直开放以来的工作，陈和沉[计算。108（1998），pp. 147-158]。引用
Implicit-explicit methods have been successfully used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work, we introduce a robust theoretical framework to analyze directly the stability and accuracy of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and provide a rigorous stability and convergence analysis for the original semi-discrete scheme under certain time step constraints. These settle several questions which have been open since the work of Chen and Shen [Comput. Phys. Comm. 108 (1998), pp. 147–158]. References