Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy
Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy
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DOI：
10.1137/110822839
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Jie Shen;Cheng Wang;Xiaoming Wang;S. Wise
中科院分区：
文献类型：
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作者：
Jie Shen;Cheng Wang;Xiaoming Wang;S. Wise
我们构造了能量为$\int_\Omega（F（\nabla\phi（{\bfx}））+ \frac{\epsilon ^2}{2}的梯度流的无条件稳定、无条件唯一可解和二阶精度（时间）格式|\Delta\phi（{\bf x}）|^2）d{\bf x}$.格式的构造涉及两个经典思想的适当组合和扩展：（i）能量泛函的适当凹凸分解和（ii）割线方法。作为一个应用，我们推导出了具有斜率选择（$F（{\bf y}）= \frac14（|{\bf y}|^2-1）^2$）或不选择斜率（$F（{\bf y}）=-\frac12\ln（1+|{\bf y}|^2）$）。给出了两类无条件稳定的唯一可解二阶格式。第一种类型继承了原始连续时间梯度流的变分结构，而第二种类型不保持变分结构。我们提出了数值模拟的情况下，与斜率的选择，验证了众所周知的物理标度律的长时间粗化过程。
We construct unconditionally stable, unconditionally uniquely solvable, and second-order accurate (in time) schemes for gradient flows with energy of the form $\int_\Omega ( F(\nabla\phi({\bf x})) + \frac{\epsilon^2}{2}|\Delta\phi({\bf x})|^2 ) d{\bf x}$. The construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional and (ii) the secant method. As an application, we derive schemes for epitaxial growth models with slope selection ($F({\bf y})= \frac14(|{\bf y}|^2-1)^2$) or without slope selection ($F({\bf y})=-\frac12\ln(1+|{\bf y}|^2)$). Two types of unconditionally stable uniquely solvable second-order schemes are presented. The first type inherits the variational structure of the original continuous-in-time gradient flow, while the second type does not preserve the variational structure. We present numerical simulations for the case with slope selection which verify well-known physical scaling laws for the long time coarsening process.