Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method
Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method
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DOI：
10.1016/j.jcp.2007.04.020
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发表时间：
2007-09
期刊：
影响因子：
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通讯作者：
S. Wise;Junseok Kim;J. Lowengrub
中科院分区：
文献类型：
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作者：
S. Wise;Junseok Kim;J. Lowengrub
我们提出了有效的，二阶精度和自适应有限差分方法来解决正则化，强各向异性的Cahn-Hilliard方程在二维和三维。当表面能各向异性足够强时，扩散界面解的平衡能级曲线中存在方向缺失，对应于尖锐界面Wulff形状中的方向缺失，各向异性Cahn-Hilliard方程变得不适定.为了正则化方程，将高阶导数项添加到能量中。这导致一个六阶，非线性抛物方程的序参数。隐式时间离散化被用来消除高阶时间步长的稳定性约束。动态块结构笛卡尔网格细化用于高度解析窄界面层。采用多层非线性多重网格方法求解隐式时间层的非线性方程组。该方法成功的关键之一是各向异性项的处理。该项在空间上以守恒形式离散，在时间上完全隐式离散。数值模拟证实了该方案的准确性，效率和稳定性。我们研究了强各向异性下的界面的动力学，并比较近平衡扩散界面的解决方案，在2D和3D的尖锐的接口武尔夫形状。我们还模拟了大规模粗化的波纹表面（在3D）的各向异性表面扩散的演变。我们展示了在粗化过程中长程有序的出现和有序粗化的有趣机制。
We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn–Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn–Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.