论文标题
Cahn-Hilliard方程的隐式解释离散化的稳定性分析
Stability analysis for the Implicit-Explicit discretization of the Cahn-Hilliard equation
论文作者
论文摘要
隐式解释方法已被广泛用于对相位场问题（例如Cahn-Hilliard方程或薄膜类型方程）的有效数值模拟。由于缺乏由小耗散系数的影响引起的最大原理和刚度，大多数现有的理论分析依赖于添加其他稳定项，微调非线性或辅助变量隐含地改变了问题的结构或以一种微妙的方式更改问题的结构或交易的准确性。在这项工作中，我们引入了一个强大的理论框架，以直接分析标准隐式解释方法的稳定性，而无需稳定或任何其他修改。我们接受 Cahn-Hilliard方程作为模型案例，并在自然时间步长约束下证明了能量稳定性，这些稳定性相对于能量缩放是最佳的。这些解决了自Chen and Shen \ cite {CS98}的工作以来已经打开的几个问题。
Implicit-Explicit methods have been widely 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 of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and prove energy stability under natural time step constraints which are optimal with respect to energy scaling. These settle several questions which have been open since the work of Chen and Shen \cite{CS98}.
