论文标题
具有动态边界条件的CAHN-HILLIARD方程的二阶准确结构保护方案
A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition
论文作者
论文摘要
我们使用离散的变分导数方法（DVDM）提出了具有动态边界条件的CAHN-HILLIARD方程的结构保存有限差方案。在这种方法中，如何离散表征方程的能量是重要且至关重要的。通过修改常规方式并使用适当的逐件求和公式，我们可以使用标准的中央差异操作员作为在该方案的离散边界条件上的外向正常衍生物的近似值。我们表明，我们所提出的方案在空间上是二阶准确的，尽管福川 - 尤西卡瓦 - 瓦达（Fukao-Yoshikawa-Wada）先前的结构披露方案（Commun。PureAppl。Anal。16（2017），1915- 1938年）在太空中是一阶准确的。同样，我们还显示了提出的方案解决方案的稳定性，存在和独特性。计算示例证明了所提出的方案的有效性。特别是通过计算示例，我们确认我们所提出的方案可以稳定地获得数值解决方案。
We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada (Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for the proposed scheme. Computation examples demonstrate the effectiveness of the proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.
