在本项目中，我们研究自由流和多孔介质流耦合问题的投影方法。利用投影格式，把自由流(Stokes/Navier-Stokes) 方程分解为几个抛物或者椭圆方程， 多孔介质流(Darcy) 方程利用时间分裂格式进行求解。同时，耦合边界条件利用上一层时间或者以求得的解来确定，以达到解耦的目的。空间离散可以利用有限元、有限体积或差分方法进行离散，给出若干稳定性好、适应性强、收敛的高精度格式。对格式进行稳定性分析和误差估计，以保证我们的数值格式是稳定的，并且具有良好的收敛性和鲁棒性。分析算法的长时间稳定性，以保证算法能进行长时间数值模拟也是稳定的。对不同雷诺数和渗透率张量进行模拟对比，以研究解的性态。同时进行长时间数值模拟，研究解的长时间动力学行为，进而验证算法的长时间稳定性。为非线性科学研究提供新的研究工具，也能更好地为计算流体力学在工程中的应用提供新的工具和理论。

In this program, we study the projection method of the coupling of the fluid flow and Darcy flow. Using the projection method, the Stokes or Navier-Stokes equations is decomposed into several parabolic or elliptic equation, the Darcy equation is solved using the time splitting scheme. Meanwhile, the coupling boundary conditions is given by the last time level, to achieve the purpose of decoupling. We use the finite element, finite volume or finite difference method to discretize the space. In this way, we will give some stability, robustness and high convergence precision numercial methods. We will give the stability analysis and error estimates of the numerical methods, to ensure that our numerical schemes are stable and has good convergence and robustness. Long time stability of our algorithms will be given, to ensure the our methods are stable for long time numerical simulation algorithms. We will give the numerical results of different Reynolds number and fluid hydraulic conductiyity tensor, and compare the resutls, to study the properties of the solutions. At the same time, long-time numerical simulations will be given to study the dynamic behavior of the solution, and then verify the long time stabilities of the algorithms. Thus, we will provide new research tools for the non-linear science, and give new tools and theories for the computational fluid dynamics in the engineering.