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.

我们开发了一种新的拉格朗日方法-流动动力学方法，以有效地捕获Allen-Cahn型方程式中的界面。这种方法的基本原理是能量变化方法（EnVarA），该方法由Rayleigh和Onsager [27]，[28]提出。与欧拉坐标系中的数值方法相比，它的主要优点是可以在拉格朗日坐标系中很少的点处有效地捕获薄界面。我们集中于一维情况，并为拉格朗日坐标中的轨迹方程构造了数值方案，该方案服从变化结构，因此是耗能的。提供了足够的数值结果，表明使用我们的流动力学方法，只有很少的点足以解析非常薄的界面。