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

多尺度和多物理场过程的数学建模和数值模拟涉及到大量的科学和工程问题。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送的建模。与不同的地质层和裂缝或不同的物理过程相关的时空尺度可能有几个数量级的变化。本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型计算方法，提高数值技术在裂缝性多孔介质应用中的效率。这些方法促进了不同模型的耦合，并允许在计算域的不同区域使用不同的时间步长和空间网格尺寸。因此，该方法可作为求解地下水流动和污染物输送、水力压裂、核废料地质处置和地质固碳等不同应用领域的大规模、强非均质、耦合演化偏微分方程的高效、准确的计算工具。在这个项目中进行的数值模拟也将为了解地质核废料储存库的长期行为和性能提供新的见解。研究生将参与这个项目，并将提供一个很好的机会，参与跨学科的研究环境。虽然区域分解方法在许多科学和工程问题上已经得到了很好的研究，但对于局部时间步进的裂缝性多孔介质应用还没有足够的重视和工作。该项目侧重于设计和分析简化裂缝模型的有效全局时域分解方法，其中将裂缝视为小于介质的一维流形。将分别考虑三个模型问题：线性输运问题、多物理场流和不可压缩两相流。所开发的方法要么基于物理传输条件，要么基于时空界面裂缝的优化传输条件；后一种情况涉及更一般的传输算子，由潜在问题的物理性质驱动，带有一些可以优化的系数，以提高迭代的收敛速度。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。该项目还将研究所提出的方法在核废料地质处置框架下裂缝性多孔介质中流体流动和污染物运移的数值模拟和调查中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.