A proximal-gradient algorithm for crystal surface evolution
A proximal-gradient algorithm for crystal surface evolution
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DOI：
10.1007/s00211-022-01320-0
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发表时间：
2020-06
影响因子：
2.1
通讯作者：
Katy Craig;Jianguo Liu;Jianfeng Lu;J. Marzuola;Li Wang
中科院分区：
文献类型：
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作者：
Katy Craig;Jianguo Liu;Jianfeng Lu;J. Marzuola;Li Wang
最近的晶体表面演化数值方法很好地符合微观动力学，但存在明显的刚度，无法在精细空间网格上进行模拟，作为对应物，我们开发了一种基于宏观偏微分方程的新的数值方法，利用其形式结构作为总变能的梯度流，相对于加权范数。这种梯度流结构与最近的几个度量空间梯度流有关，包括2-Wasserstein流及其对非线性流动的推广。我们开发了一种新的梯度流的半隐式时间离散化，灵感来自经典的最小化运动方案（在2-Wasserstein情况下称为JKO方案）。然后，我们使用原始对偶混合梯度（PDHG）方法来计算半隐式格式的每个元素。在一维情况下，在可积性及其倒数的一般可积性假设下，证明了PDHG方法对半隐格式的收敛性。最后，通过采用我们的PDHG方法的有限差分近似，我们得到了一个完全离散的数值算法，其迭代以独立于空间离散化的速率收敛：特别是，收敛性质不会随着我们改进空间网格而恶化。最后，我们用几个数值例子来说明我们的方法的特性，包括在局部最大值处形成facet，在局部最小值处固定，以及随着空间和时间离散化的改进而收敛。
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weightednorm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.