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。

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.