In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.
翻译:在本文中,我们提议并分析一个第一阶和第二阶时间步骤方案,用于分析厌同相相位阶段的登地晶晶体生长模型。拟议办法基于艾伦-卡恩方程式的辅助变量办法和对艾伦-卡恩方程式和温度方程式结合的术语的微妙处理。前者的构想是引入适当的辅助变量,以便利为大型梯度流建立高阶稳定方案。我们提出了一种新的技术,处理晶体生长模型所涉的混合条件,并引入适当的稳定条件,从而形成完全脱钩的组合计划,既满足离散能源法,又不影响汇合顺序。一个微妙的实施表明,拟议的方案可以非常高效地实现。也就是说,只需要在每一阶段解决四个线性椭圆方程式和简单的代数方程式。与现有方案进行详细比较,并强调新办法的优势。据我们所知,这是第一个完全脱钩、线性、无条件稳定的第二阶次方案,可以与可变形晶体生长模型和可变性参数完全脱钩。