An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem
An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem
复制标题
DOI：
10.1016/j.jcp.2020.109521
复制
发表时间：
2019-04
期刊：
影响因子：
--
通讯作者：
Xiaoyu Wei;Shidong Jiang;A. Klöckner;Xiaoping Wang
中科院分区：
文献类型：
--
作者：
Xiaoyu Wei;Shidong Jiang;A. Klöckner;Xiaoping Wang
我们提出了一个积分方程的方法来解决Cahn-Hilliard方程配备边界条件，模型固体表面与规定的杨氏角。系统在时间上的离散化，使用凸分裂导致在每个时间步修改的双调和方程。为了解决这个问题，我们分裂成一个自由空间内核计算的体积势的解决方案，加上第二类积分方程（SKIE）。体积潜力的评估与基于盒的体积FMM方法的帮助下。对于非箱形区域，通过求解一个双调和Dirichlet问题来扩展源密度。近奇异边界积分的计算使用正交扩展（QBX）与FMM加速。我们的方法在表面/体积自由度的数量上具有线性复杂性，并且可以通过自适应细化来管理函数扩展的误差，从而在空间中实现高阶收敛。
We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, the source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension.