基于位势理论求解粘性系数间断的两相Navier-Stokes方程的交错网格法

粘性系数间断的两相Navier-Stokes(NS)方程常用于模拟流体力学、生物学、材料科学等许多问题。粘性系数间断导致的急剧界面问题为高效高精度地求解两相NS方程提出了许多挑战。本项目将在交错网格上研究粘性系数间断的两相NS问题的高效数值方法及其误差估计。首先本项目打算基于位势理论将粘性系数间断的两相NS界面问题或NS半离散后所得Stokes界面问题的求解简化为单相情形，并设计对应鞍点系统的快速求解方法。其次，发展时空一致四阶精度的数值格式，并结合界面捕捉方法，实现移动界面问题的数值模拟。最后，对单相Stokes/NS界面问题的MAC格式，建立离散LBB条件，并给出数值格式的稳定性，进而探讨其离散能量不等式与最优误差估计。本项目的研究将应用于某些大规模三维流体界面问题的数值模拟，如Stoke/NS-Darcy模型等。

The two-phase Navier-Stokes (NS) equations with discontinuous viscosity coefficient are often used to simulate many problems such as hydrodynamics, biology, materials science, etc. The discontinuity of viscosity coefficient will lead to sharp interface problems, so how to simulate such problems with high efficiency and accuracy faces many challenges. This project is aimed to study some efficient numerical methods to simulate the two-phase NS equations with discontinuous viscosity under staggered grids. Firstly, based on the potential theory, the two-phase NS interface problem with discontinuous viscosity coefficient or Stokes interface problem after NS semi discretization are simplified to the single case, and the fast solver for the saddle point systems is designed. Secondly, an efficient numerical scheme with fourth-order accuracy both in time and space is developed, Also the moving interface problem is simulated with some interface tracking methods. Finally, for the MAC scheme of the single-phase Stokes/NS interface problem, the discrete LBB condition is established, and the stability of the numerical scheme is given, then the discrete energy inequality and the optimal error estimation are discussed. The research of this project will be applied to the numerical simulation of some three-dimensional fluid interface problems with large-scale, such as the Stokes/NS-Darcy model.