Non-Iterative Multi-Physics Domain Decomposition Method for the Navier-Stokes-Darcy Model

Navier-Stokes-Darcy模型对于准确描述许多重要的现实世界问题至关重要，例如孔洞/裂缝多孔介质中的流动问题、喀斯特含水层中的地下水系统、地表和地下流动之间的相互作用以及工业过滤。数值方法求解该模型的效率对于大规模应用具有重要意义。但是，大多数已发表的相关作品都集中在一个更简单的模型上，由于它没有考虑几个关键的现实因素，因此对许多应用来说并不准确。本课题旨在研究更为现实的Navier-Stokes-Darcy模型，并设计一种有效的并行求解方法。该项目的完成将为流体耦合流动和多孔介质流动的研究提供很大的进展。本课题的基本思想和框架有可能应用于其他有趣的多物理场耦合问题。该项目为本科生和研究生提供了许多宝贵的培训机会，包括开发新的方法和软件包、数学分析和工程应用。正在开发的方法将应用于模拟孔洞/裂缝多孔介质中的流动。研究人员计划通过在专业会议和座谈会上展示工作成果，并组织特别会议，将方法和软件包传播给更多的工程师和科学家。本项目的目标是对具有Beavers-Joseph （BJ）界面条件和缺陷边界条件的Navier-Stokes-Darcy模型的非迭代多物理场域分解方法的开发、分析和应用进行系统研究。所提出的非迭代多物理场域分解方法能够有效地同时处理由于上述关键现实因素之间的相互作用而产生的重大困难。其核心思想包括直接预测界面的非迭代算法、由BJ条件等三种物理界面条件产生的带有Robin条件的多物理场分解以及处理缺陷边界条件的拉格朗日乘子法。主要难点是非线性平流、BJ条件、Robin-Robin分解参数的内在相互作用，以及拉格朗日乘子法引起的鞍点问题。

The Navier-Stokes-Darcy model is crucial to accurately describe many important real world problems, such as the flow problems in vuggy/fractured porous media, groundwater systems in karst aquifers, interaction between surface and subsurface flows, and industrial filtrations. The efficiency of the numerical methods to solve this model is very important for the large-scale applications. But most of the related published works focus on a simpler model, which is not accurate for many applications since it does not consider several crucial realistic factors. This project is to study a more realistic Navier-Stokes-Darcy model and design an efficient numerical method to solve it in parallel. The completion of this project will provide large advances in the study of the coupled fluid flow and porous media flow. The basic idea and framework of this project have potential to be applied to other interesting multi-physics coupling problems. This project provides undergraduate and graduate students many valuable training opportunities in the development of new methods and packages, mathematical analysis, and engineering applications. The methods under development will be applied to simulate the flow in vuggy/fractured porous media. The investigators plan to disseminate the methods and software packages to more engineers and scientists by presenting the work in professional conferences and colloquia and organizing special sessions. The goal of this project is to carry out a systematic research on the development, analysis, and application of a non-iterative multi-physics domain decomposition method for the Navier-Stokes-Darcy model with the Beavers-Joseph (BJ) interface condition and defective boundary conditions. The proposed non-iterative multi-physics domain decomposition method enables us to efficiently and simultaneously deal with the significant difficulties arising from the interaction among the aforementioned crucial realistic factors. The key ideas include a non-iterative algorithm with a direct prediction on the interface, a multi-physics decomposition with Robin conditions arising from the three physical interface conditions which include the BJ condition, and the Lagrange multiplier method to deal with the defective boundary condition. The major difficulty is the inherent interaction of the nonlinear advection, BJ condition, parameters in the Robin-Robin decomposition, and the saddle point problem arising from the Lagrange multiplier method.