Global-in-Time Domain Decomposition Methods for Evolution Partial Differential Equations with Applications to Flow and Transport in Fractured Porous Media

多尺度、多物理场过程的数学建模和数值模拟是许多科学和工程问题的本质内容。特别是在环境科学和地球科学的许多应用中，人们关注的是含有裂缝和断层的多孔介质中的流动和输运的建模。在那里，与各种地质层和断裂或不同的物理过程有关的空间和时间尺度可能有几个数量级的变化。该项目的目标是通过设计和分析基于并行全局时域分解的新计算方法来提高裂缝多孔介质应用的数值技术的效率。这些方法有利于不同模型的耦合，并使不同的时间步长和空间网格大小的计算域的不同区域中的使用。因此，所提出的方法可以用来作为一个高效和准确的计算工具，解决大规模，强异质性，耦合发展偏微分方程所产生的不同的应用领域，如地下水流动和污染物输运，水力压裂，地质处置核废料和地质固碳。在这个项目中进行的数值模拟也将提供新的见解的长期行为和性能的地质核废物处置库的理解。研究生将参与这个项目，并将提供一个很好的机会，参与一个跨学科的研究环境。虽然区域分解方法已经很好地研究了许多科学和工程问题，但没有足够的关注和工作一直致力于破裂多孔介质的应用与当地的时间步进。该项目的重点是简化裂缝模型的有效的全球时域分解方法的设计和分析，其中裂缝被视为比介质小一维的流形。将考虑三个模型问题：线性输运问题，多物理场流和不可压缩两相流，分别。所开发的方法是基于物理传输条件或优化传输条件的时空界面裂缝;后者的条件涉及更一般的传输运营商，由物理学的基本问题，一些系数，可以优化，以提高收敛速度的迭代。重要的是，所提出的方法使得在界面裂缝和周围介质中使用不同的时间步长和空间网格成为可能。该PI还将研究所提出的方法的应用，以数值模拟和调查的流体流动和污染物输运的断裂多孔介质所产生的地质核废料disposal.This奖项的框架反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Mathematical modeling and numerical simulation of multiscale and multiphysics processes are essentially involved in a large number of scientific and engineering problems. Particularly in many applications in environmental sciences and geosciences, one is concerned with the modeling of flow and transport in porous media containing fractures and faults. There the spatial and temporal scales associated with various geological layers and fractures or different physical processes may vary with several orders of magnitude. The goal of this project is to enhance the efficiency of numerical techniques for fractured porous medium applications by designing and analyzing novel computational methods based on parallel global-in-time domain decomposition. These methods facilitate the coupling of different models and enable the use of different time step sizes and spatial mesh sizes in different regions of the computational domain. Thus the proposed methods can be used as an efficient and accurate computational tool for solving large-scale, strongly heterogeneous, coupled evolution partial differential equations arising from diverse application fields such as groundwater flow and contaminant transport, hydraulic fracture, geological disposal of nuclear waste and geological carbon sequestration. The numerical simulations carried out in this project would also provide new insights to the understanding of the long-term behavior and performance of geological nuclear waste repositories. Graduate students will be involved in this project and will be offered a great opportunity to participate in an interdisciplinary research environment.Although domain decomposition methods have been well studied for many scientific and engineering problems, no enough attention and work have been devoted to fractured porous medium applications with local time stepping. This project focuses on the design and analysis of efficient global-in-time domain decomposition methods for reduced fracture models, in which the fractures are treated as manifolds of one dimension less than the medium. Three model problems will be considered: the linear transport problem, the multiphysics flow and the incompressible two-phase flow, respectively. The developed methods are based on either physical transmission conditions or optimized transmission conditions on the space-time interface fractures; the latter conditions involve more general transmission operators, motivated by the physics of the underlying problem, with some coefficients that can be optimized to improve the convergence rates of the iterations. Importantly, the proposed methods make possible the use of different time step sizes and spatial grids in the interface fractures and in the surrounding medium. The PI will also study the application of the proposed methods to numerical simulation and investigation of fluid flow and contaminant transport in fractured porous media arising from the framework of geological nuclear waste disposal.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.