Stokes-Darcy耦合问题的解耦和局部并行算法研究

Stokes-Darcy model arises in many applications such as groundwater system in karst aquifers, interaction between surface and subsurface flows and petroleum extraction. It describes a free flow of a liquid by the Stokes equation and the confined flow in a porous media by Darcy equation; the two flows are coupled through interface conditions. The difference of local models and existence of the interface condition lead to various mathematical and numerical difficulties. They make the decoupling methods for solving Stokes-Darcy coupling problem become one of the hot topics in Computational Mathematics and Computational Fluid Dynamics...Based on the ideas of two-grid method and Robin-Robin domain decomposition method, we propose the decoupling methods for (Navier-)Stokes-Darcy coupling problems. For stationary case, we use coase grid solution to decouple the problem; for the evolutionary case, by lagging the interfacial coupling terms onto the previous time level, decoupling methods separate the coupling problem into (Navier-)Stokes and Darcy problems, it reduces the computational scale and improves its efficiency, it is also suitable for today's grid computing environment because it can efficiently and effectively exploit the existing computing resources, including both hardware and software. After that, combining with parallel computational techniques, we further divide each physical-based domain into several non-overlapping subdomains. Then we solve a serial of residual equations of (Navier-)Stokes or Darcy problem on these subdomains in parallel, which achieve greater parallelism and improve its efficiency. ..Further combined with the diffusion equation, we provide a theoretical tool for the study of the propagation of contaminants in groundwater....

Stokes-Darcy耦合问题在石油工程、地下水污染等领域有重要应用。由于两区域内方程特征的差异以及交界面上耦合条件的存在，使得构造高效稳定的解耦算法求解该问题成为计算数学和计算流体力学的研究热点。..我们基于两重网格思想和Robin-Robin区域分解思想，构造(Navier-)Stokes-Darcy耦合问题的解耦算法。对于定常问题，利用粗网格解处理交界面耦合项来解耦方程，对于非定常问题，利用时间步推进构造解耦算法。实现在当前迭代步或时间步上仅求解独立的(Navier-)Stokes和Darcy问题，降低计算规模，提高计算效率，并且可以利用已有算法程序资源降低编程难度。解耦后，各自区域再分别进行分块，并行求解若干个(Navier-)Stokes或Darcy残量子问题，进一步提高解耦算法的计算效率。..耦合模型进一步结合扩散方程，为研究地下水中污染物的传播情况提供理论依据。

现实中的许多物理现象可用耦合的偏微分方程系统进行刻画。 例如Stokes-Darcy耦合问题是由自由流体区域的Stokes方程和多孔介质区域的Darcy定律结合交界面条件构成的，它在石油开采和地下水污染等领域中有着重要应用。流-流耦合模型是由两个Navier-Stokes方程结合交界面条件构成的，它在大气海洋环流和天气预测的数值模拟中有重要应用。Dual-porosity-Stokes耦合模型包括双重空隙介质中的流体流动（即双孔隙模型）以及在裂缝和管道中的流体流动（即Stokes模型），两个子模型通过四个交界面条件耦合在一起，该模型在非常规油气藏中应用广泛。本项目主要内容包括：一、针对非定常Stokes-Darcy耦合问题，提出了解耦算法并分析其最优误差估计。此外，我们还提出了在自由流体区域和多孔介质区域都采用混合稳定化有限元方法来模拟的解耦算法，这样既可以求出自由流体的速度又可以求出多孔介质区域的流体速度，这为下一步将Stokes-Darcy耦合模型结合扩散方程提供了基础。二、针对具有线性和非线性交界面条件的非定常流-流耦合模型，构造了两类解耦算法，分别是基于两重网格的解耦算法和基于时间步推进的解耦算法，并分析了各算法的稳定性和收敛性，数值上也验证算法的可行性。三、针对Dual-porosity-Stokes耦合模型，构造解耦算法。四、针对磁流体方程，构造其局部并行有限元方法，此项工作为后续研究耦合问题的局部并行算法提供了技术支持。本项目所构造的三类耦合问题的解耦算法，不仅可以为工程实验提供理论依据，还可以为其他类耦合问题的数值求解提供算法依据和理论分析框架。