In this paper, we propose and analyze an efficient numerical method for the anisotropic phase field dendritic crystal growth model, which is challenging because we are facing the nonlinear coupling and anisotropic coefficient in the model. The proposed method is a two-step scheme. In the first step, an intermediate solution is computed by using BDF schemes of order up to three for both the phase-field and heat equations. In the second step the intermediate solution is stabilized by multiplying an auxiliary variable. The key of the second step is to stabilize the overall scheme while maintaining the convergence order of the stabilized solution. In order to overcome the difficulty caused by the gradient-dependent anisotropic coefficient and the nonlinear terms, some stabilization terms are added to the BDF schemes in the first step. The second step makes use of a generalized auxiliary variable approach with relaxation. The Fourier spectral method is applied for the spatial discretization. Our analysis shows that the proposed scheme is unconditionally stable and has accuracy in time up to third order. We also provide a sophisticated implementation showing that the computational complexity of our schemes is equivalent to solving two linear equations and some algebraic equations. To the best of our knowledge, this is the cheapest unconditionally stable schemes reported in the literature. Some numerical examples are given to verify the efficiency of the proposed method.

翻译：本文提出并分析了一种针对各向异性相场树枝晶生长模型的高效数值方法。该模型的挑战在于其非线性耦合及各向异性系数。所提方法为两步格式：第一步，通过使用最高三阶的BDF格式求解相场方程和热传导方程，计算中间解；第二步，引入辅助变量对中间解进行稳定化处理。该方法的关键在于通过稳定化处理维持整体格式的收敛阶，同时确保数值稳定性。为克服各向异性系数（依赖梯度项）及非线性项带来的计算困难，第一步的BDF格式中加入了稳定化项。第二步采用带松弛机制的广义辅助变量方法。空间离散采用傅里叶谱方法。理论分析表明，该格式无条件稳定，且时间精度可达三阶。我们还给出了精巧的实现方案，证明该方法的计算复杂度等价于求解两个线性方程及若干代数方程。据我们所知，这是文献中计算代价最低的无条件稳定格式。最后通过数值算例验证了该方法的有效性。