SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1541-B1569, January 2020.
In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states of the original Allen--Cahn type model. Our discretization is based on a prescribed energy-dissipation law in terms of the flow map. By employing a discrete energetic variational approach, this scheme preserves the variational structure of the continuous energy-dissipation law and is energy stable. Plentiful numerical tests show that, by choosing the initial value properly, our method can compute the desired equilibrium state and capture the thin diffuse interface with a small number of mesh points.

中文翻译：

---

相场模型的变分拉格朗日方案：一种离散的能量变分方法

SIAM科学计算杂志，第42卷，第6期，第B1541-B1569页，2020
年1月。在本文中，我们提出了一种用于修正相场模型的变分拉格朗日方案，该方案可以计算原始Allen-- Cahn型模型。我们的离散化是根据流程图根据规定的能量耗散定律进行的。通过采用离散的能量变分方法，该方案保留了连续能量耗散定律的变化结构，并且能量稳定。大量的数值测试表明，通过适当地选择初始值，我们的方法可以计算出所需的平衡状态，并捕获带有少量网格点的薄扩散界面。