Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model
Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model
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DOI：
10.1016/j.ijheatmasstransfer.2021.121750
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发表时间：
2021-12
影响因子：
5.2
通讯作者：
Xiaofeng Yang
中科院分区：
文献类型：
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作者：
Xiaofeng Yang
在这项工作中，我们考虑的各向异性相场枝晶生长模型，这是一个高度复杂的耦合非线性系统的各向异性Allen-Cahn方程和热方程的数值近似。通过结合一种新的显式辅助变量IEQ方法的时间离散和空间离散的谱Galerkin方法，我们开发了第一个完全离散的数值格式与线性，解耦的结构，无条件的能量稳定性，和二阶时间精度的特定相场枝晶模型。在获得完全解耦结构和保持能量稳定的过程中，两个辅助变量的引入和两个辅助常微分方程的设计起着至关重要的作用。该方法在每一时间步只需要求解几个常系数椭圆方程组，因此具有较高的计算效率。严格证明了该格式的无条件能量稳定性，并给出了具体的实现过程。通过对二维和三维枝晶生长的数值模拟，进一步验证了算法的收敛速度、能量稳定性和有效性。
In this work, we consider numerical approximations of the anisotropic phase-field dendritic crystal growth model, which is a highly complex coupled nonlinear system consisting of the anisotropic Allen-Cahn equation and the heat equation. Through the combination of a novel explicit auxiliary variable IEQ approach for temporal discretization and the spectral-Galerkin approach for spatial discretization, we develop the first fully-discrete numerical scheme with linearity, decoupled structure, unconditional energy stability, and second-order time accuracy for the particular phase-field dendritic model. In the process of obtaining a full decoupling structure and maintaining energy stability, the introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role. The designed scheme is highly efficient because only a few elliptic equations with constant coefficients are needed to be solved at each time step. The unconditional energy stability of the scheme has been strictly proved, and the detailed implementation process is given. Through several numerical simulations of 2D and 3D dendritic crystal growth examples, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.