A new interface capturing method for Allen-Cahn type equations based on a flow dynamic approach in Lagrangian coordinates, I. One-dimensional case
A new interface capturing method for Allen-Cahn type equations based on a flow dynamic approach in Lagrangian coordinates, I. One-dimensional case
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一种新的基于拉格朗日坐标下流动动力学方法的 Allen-Cahn 型方程界面捕获方法，I. 一维情况
DOI：
10.1016/j.jcp.2020.109509
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发表时间：
2020
影响因子：
4.1
通讯作者：
Shen, Jie
中科院分区：
文献类型：
--
作者：
Cheng, Qing;Liu, Chun;Shen, Jie
我们开发了一种新的拉格朗日方法-流动动力学方法，以有效地捕捉界面的Allen-Cahn型方程。这种方法的基本原理是能量变分方法（EnVarA），由Rayleigh和Onsager [27]，[28]激发。与欧拉坐标系下的数值方法相比，该方法的主要优点是可以在拉格朗日坐标系下用较少的点有效地捕捉薄界面。我们集中在一维的情况下，并构造数值方案的轨迹方程在拉格朗日坐标，服从变分结构，因此，是能量耗散。充分的数值结果表明，只有较少的点是足够的，以解决非常薄的接口，使用我们的流动动力学方法。
We develop a new Lagrangian approach — flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager [27], [28]. Its main advantage, comparing with numerical methods in Eulerian coordinates, is that thin interfaces can be effectively captured with few points in the Lagrangian coordinate. We concentrate in the one-dimensional case and construct numerical schemes for the trajectory equation in Lagrangian coordinate that obey the variational structures, and as a consequence, are energy dissipative. Ample numerical results are provided to show that only fewer points are enough to resolve very thin interfaces by using our flow dynamic approach.