On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model
On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model
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DOI：
10.1002/nme.6697
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发表时间：
2021-04
影响因子：
2.9
通讯作者：
Xiaofeng Yang
中科院分区：
文献类型：
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作者：
Xiaofeng Yang
各向异性相场枝晶生长模型是将各向异性Allen-Cahn方程和热方程耦合在一起的高度非线性系统。由于系统的高度各向异性和非线性耦合，如何发展一个准确和有效的，特别是一个完全解耦的格式，一直是一个具有挑战性的问题。为了解决这一挑战，在本文中，我们构造了一种新的完全解耦的数值格式，该格式也是线性的，能量稳定的，并且具有二阶时间精度。实现完全解耦结构的关键思想是引入一个常微分方程来处理满足所谓“零能量贡献”性质的非线性耦合项。该格式在每一时间步只需要求解几个完全解耦的常系数椭圆方程，因而非常有效且易于实现。我们严格证明了每一步的可解性和无条件能量稳定性，并进行了大量的二维和三维数值模拟，以证明其稳定性和数值精度。
The anisotropic phase‐field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen–Cahn equation and the thermal equation together. Due to the high anisotropy and nonlinear couplings in the system, how to develop an accurate and efficient, especially a fully decoupled scheme, has always been a challenging problem. To solve the challenge, in this article, we construct a novel fully decoupled numerical scheme which is also linear, energy stable, and second‐order time accurate. The key idea to realize the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms satisfying the so‐called “zero‐energy‐contribution” property. This scheme is very effective and easy to implement since only a few fully decoupled elliptic equations with constant coefficients need to be solved at each time step. We rigorously prove the solvability of each step and the unconditional energy stability, and perform a large number of numerical simulations in 2D and 3D to demonstrate its stability and accuracy numerically.