非稳态Navier-Stokes/Darcy耦合问题的解耦算法研究

The coupling of Navier-Stokes and Darcy problem is one of the hot topics Computational Fluid Dynamics due to its importance in modeling problems such as surface fluid flow coupled with flow in a porous media. The mixture of the coupling model leads to various mathematical and numerical difficulties. For instance, interface coupling conditions involve different control variables from different local models and may have complex, or even nonlinear forms. Direct numerical simulation usually requires very fine computational meshes, and it's still a big challenge for solving incompressible flows even in the highly-developed computer science nowadays..By separating the coupling problem into Navier-Stokes and Darcy problems, decoupling method reduces the computational scale and improves its efficiency, it is also suitable for today's grid computing environment because it can efficiently and effictively exploit the existing computing resources, including both hardware and software. . .For non-stationary Navier-Stokes/Darcy coupling system, we consider the fully discretization, by simply lagging the interfacial coupling terms onto the previous time levels, we can separate the fluid and the porous media parts and solve them individually. Meanwhile, combining with different time discretization schemes，we develop different decoupling methods and prove their stabilities and convergent properties theoretically. We also verify the efficiencies and feasibilities of the methods from numerical experiments.

利用解耦算法求解不可压缩流体（Navier-Stokes方程）和多孔介质流体（Darcy方程）的耦合问题是计算流体力学中的一个重大课题。直接数值求解耦合问题存在求解规模巨大、多模型混合、交界面处理复杂、计算资源有限等难点。. 解耦算法是通过恰当方式解除耦合问题在交界面上的耦合限制，进而将其分离成两个相对独立的Navier-Stokes和Darcy问题并分别求解，以达到降低计算规模、提高计算效率并充分利用已有算法程序资源的目的。. 针对非稳态Navier-Stokes/Darcy耦和问题，进行时空全离散时，利用在每个时间层上显式处理交界面上耦合项信息以达到解耦目的。结合不同的时间离散格式构造不同的解耦算法，并从理论上分析其各自的稳定性和收敛性。同时，通过数值实验，验证算法的高效性和可行性。

针对多模型耦合问题，我们的研究侧重于解耦算法的构造。针对两个扩散方程的耦合问题，提出两类二阶解耦算法：BDF2格式和AMB2格式，分别从理论上和数值上验证了二者的稳定性、收敛性和高效性【附件1】。将其推广到大气-海洋耦合模型，我们已经完成了解耦算法的数值模拟部分，正在进行理论分析方面的工作。具有Oldroyd B型本构方程的非定常粘弹性流问题是典型的耦合问题，我们提出了间断Galerkin有限元方法，从理论上证明了我们算法是无条件收敛的【附件2】。我们将解耦思想应用到该模型上，使系统中耦合着的两个方程分开求解，进而降低计算规模，提高计算效率， 这部分工作正在最后的数值模拟和整理当中。 针对磁流体方程，我们提出了Crank-Nicolson外推格式，在关于初始条件和右端项的假设前提下证明了算法的无条件收敛性【附件3】。该项目还支持了局部并行有限元方面的工作【附件4】，这是我们在将耦合问题解耦之后的下一步研究计划，将并行技术引入到解耦算法中，进一步提高计算效率。此外，项目组成员还进行了关于不定规划和二次规划相关的工作【附件5,6】。