多相相场模型保结构算值算法研究

Multi-phase phase field model is widely used in solidification theory system, micro-structure analysis of industrial alloys, fluid mechanics，life sciences，and other fields. Moreover, it is very difficult to construct some decoupled, linear and unconditional energy stability numerical methods due to the high coupling and nonlinear term of multi-phase field model, and it is also difficult to analyze numerically. We hope to develop theoretical achievements in this field through the study of this kind of problems. The contents include: ①Some linear, completely decoupled and non-iterative numerical methods are constructed to solve the two-phase flow phase field model and the three-phase Cahn-Hilliard phase field model. In addition, the newly constructed numerical method can maintain long-term stability; ②The unconditional energy stability, uniqueness and error estimation of these schemes are proved based on S-SAV and predictive correction SAV method.

多相相场模型不论是在凝固理论体系、还在是工业合金的微观组织分析、流体力学和生命科学等领域中都有着广泛的应用。而且，多相相场模型由于高度耦合和非线性项使得构造解耦、线性、无条件能量稳定的数值方法相当困难，而且数值分析难度大。我们希望通过对该类问题的研究，发展这方面的理论成果。项目主要研究内容包括：①构造一些线性、完全解耦、非迭代的数值方法来求解两相流相场模型和三相Cahn-Hilliard相场模型，此外，新构造的数值方法能够保持长时间稳定性；②证明这些格式基于S-SAV和预估校正的SAV方法的无条件能量稳定性、唯一性和误差估计。

多相相场模型不论是在凝固理论体系、还在是工业合金的微观组织分析、流体力学和生命科学等领域中都有着广泛的应用。多相相场模型由于高度耦合和非线性项使得构造解耦、线性、无条件能量稳定的数值方法相当困难，而且数值分析难度大。为了解决上述难题，本项目围绕几类相场模型构造保结构数值算法，主要研究结果包括：1)对Gray-Scott模型，构造了一个二阶、线性显格式，证明了格式的二阶精度，得到了相应的误差估计；2)对Molecular Beam Epitaxial模型，构造了一个带稳定子的保结构算法，证明格式的无条件能量稳定性；3)对Navier-Stokes 耦合多元Cahn-Hilliard模型，基于EIEQ方法，构造了一个时间二阶、解耦和无条件能量稳定的数值方法；4)构造了各项异性的Cahn-Hilliard模型任意高阶算法，证明了无条件能量稳定性。以上述研究结果为基础，发表了SCI论文三篇。