Stokes和Navier-Stokes方程及相关流动问题的交错网格MAC方法研究

The MAC (Karker-and-Cell) finite difference method for Stokes equation is a simple, practical and classical method. Numerical calculations show that the method has super convergence property, but the corresponding theoretical analysis is not perfect, which restrict its applications. This project do researches into the staggered grid MAC finite difference methods based on non-uniform mesh for steady and unsteady Stokes equations, Navier-Stokes equations, and the related flow problems, for instance, the Darcy-Stokes-Brinkman problem, the coupled Stokes (or N-S) and Darcy (or Darcy-Forchheimer) flow problem. The constructed MAC methods should be able to keep the conservation of momentum and mass respectively, they should also be able to keep divergence free and other properties. We will give the analysis for the various methods of stability for the pressure and velocity. For each method we will try to find the auxiliary functions of pressure and velocity, by which we will establish the superconvergence properties. By numerical experiments we will analyze the advantages and disadvantages of the methods. Besides the study of staggered grid MAC methods based on rectangular partition we will do further researches on staggered grid MAC methods based on triangulation. We will also do research on the staggered grid algorithm for the following problems: The system for multi-component fluid flow, thermal fluid flow problem including the equations to describe the pressure and the velocity and, moreover, including a nonlinear convection diffusion equation to description the component mass conservation or energy conservation. We will give the complete stability and superconvergence analysis.

Stokes方程的MAC（Karker-and-Cell）差分方法是一类简单实用的经典方法，计算发现有超收敛性，但理论研究不完善，制约了该方法的应用。本项目研究稳态和非稳态Stokes方程、N-S方程、以及相关的流动问题如Darcy-Stokes-Brinkman方程、耦合的Stokes（或N-S）与Darcy（或Darcy-Forchheimer）流等的非一致剖分交错网格MAC差分方法，构造的MAC方法要保持动量守恒、质量守恒等性质；分析各类方法的关于压力和速度的稳定性；对每一算法寻找辅助的压力和速度函数，借助它们建立超收敛性；通过数值实验分析算法的实用性。在研究矩形剖分MAC方法基础上进一步研究三角形剖分的交错网格MAC算法。多组份流体、热流体等流动问题除包含压力和速度方程外，还包含描述组份质量或能量守恒的非线性对流扩散方程，研究这些问题的交错网格算法，分析稳定性和超收敛性。

Stokes方程的MAC差分方法或有限体积方法是计算流体力学一类简单实用的经典方法，数值计算发现算法具有超收敛性，但缺乏理论证明，制约了该方法的应用。本项目的主要研究成果如下。我们通过构造压力和速度的离散辅助函数，证明了稳态、非稳态和、时间方向分数阶的Stokes方程MAC算法的速度及其各项一阶偏导数、压力关于L2范数的若干超收敛性，并提出后处理技术来提高速度的计算精度；对非稳态N-S方程构造了特征线MAC算法并证明了其超收敛性，这些理论成果都通过数值试验得到了验证。在此基础上对Darcy-Stokes-Brinkman方程、耦合的Stokes与Darcy流动问题等构造了非一致剖分交错网格MAC差分方法、混合元算法，对含有多种组分的多孔介质蚓孔酸化模型构造出交错网格MAC方法，并证明这些算法的超收敛性、保持质量守恒等性质。另外我们研究构造了弹性体和多孔弹性体的交错网格差分算法和混合元算法、分数阶微分方程的MAC算法等多种数值算法，证明这些方法的的稳定性、收敛性以及超收敛性。并对各类算法通过数值实验分析算法的有效性和实用性。本项目资助发表论文41篇，包括发表在SIAM J Numer Anal（2017,2018）、Math Comp（2019）、J Fluid Mech（2019）、Computer Mechods Appl Mech Engrg（2017）等顶级期刊上的论文。本项目资助培养博士生10人，已毕业6人，其中两人毕业后获得国家博士后创新人才计划支持、一人在学期间获CSIAM年会优秀学生论文奖。