Fully Locally Conservative Characteristic Methods for Transport Problems

化学示踪剂在环境流体(如地下水中的污染物)中的传输可以用欧拉-拉格朗日数值方法来近似。这些方法在局部保存示踪剂质量，但不保存环境流体的质量。因此，它们计算的密度不准确，随着时间的推移，这可能会严重降低溶液的质量。PI和他的同事最近定义了体积校正特征-混合方法(VCCMM)来解决最简单的运输问题，将运输问题视为描述示踪剂和环境流体运动的两个方程组，每个方程都必须局部守恒。该项目将：(1)完成VCCMM的开发，将其置于坚实的理论基础上，并开发软件的并行版本；(2)改进VCCMM并将其推广到更复杂的流动；(3)开发网格适应版本；以及(4)开发求解非线性输运问题的方法，包括可混相、可压缩流动和两相非混相流动，其解可能包含激波和稀疏面。由于离散近似的数值扩散较少，并保留了重要的物理原理，因此该项目有望在长时间模拟的运输问题的近似方面取得显著改进。该项目预计将产生更广泛的影响，包括：(1)开发可应用于广泛实际问题的科学软件工具；(2)在多学科环境中培训一名博士生；(3)通过允许更好地对地质盆地形成、长寿命放射性同位素衰变、混合指进和两相流进行更好的建模，从而产生社会效益。预测一种化学物种(称为示踪剂)在另一种环境流体中的运动的能力在许多应用中都很重要。例如，在地下水污染物迁移研究中出现了这种需要。本项目研究如何通过计算机模拟来改进示踪剂输送的预测。最先进的拉格朗日型数值算法通过显式计算单个粒子在小空间区域内的运动来模拟示踪剂的传输。示踪质量是守恒的，这意味着数值计算不会人为地创建或销毁任何质量。对于涉及例如污染物的研究来说，这是一个关键的性质，因为即使是很小的浓度也可能对人体有毒，示踪剂的任何产生或降解都必须是由于物理和化学过程而不是数字人工制品。然而，拉格朗日方法并不守恒周围流体的质量。这导致示踪剂密度不准确。也就是说，尽管示踪剂质量是守恒的，但其浓度计算不正确，这可能导致反应动力学中的严重不准确，以及随着时间的推移预测运动的降级。PI采取的解决这些困难的方法是同时考虑示踪剂和环境流体的传输，其中每一种都必须保守。这项研究有望显著提高长时间模拟运输问题的近似性，并在多学科环境中培训至少一名博士生。这项工作具有潜在的社会效益，适用于地下水污染、石油和天然气生产以及二氧化碳封存等问题。

The transport of a chemical tracer within an ambient fluid (such as a contaminant in groundwater), can be approximated by Eulerian-Lagrangian numerical methods. These methods conserve tracer mass locally, but not the mass of the ambient fluid. They therefore compute inaccurate densities, which can seriously degrade the quality of the solution over time. The PI and coworker recently defined the Volume Corrected Characteristics-Mixed Method (VCCMM) for the simplest transport problem, by considering the transport problem not as a single hyperbolic equation for the tracer, but rather as a system of two equations describing the motion of both the tracer and ambient fluids, each of which must be conserved locally. This project will:(1) Complete the development of VCCMM, by placing it on a sound theoretical footing and developing a parallel version of the software; (2) Improve and extend VCCMM to more complex flows; (3) Develop a grid adapted version; and (4) Develop methods for nonlinear transport problems, including miscible, compressible flows, and two-phase, immiscible flows, for which the solution may contain shocks and rarefactions. The project is expected to result in significant improvement in the approximation of transport problems for long time simulation, because the discrete approximation will have less numerical diffusion and preserve important physical principles. The project is expected to have broader impacts, including: (1) Developing a scientific software tool that can be applied to a wide range of practical problems; (2) The training of one Ph.D. student in a multidisciplinary environment; and (3) Societal benefits by allowing better modeling of, e.g., geologic basin formation, long-lived radio-isotope decay, miscible fingering, and two-phase flows.The ability to predict the movement of a chemical specie, called a tracer, within another, ambient fluid is important in many applications. For example, the need arises in ground-water contaminant migration studies. This project investigates ways to improve the prediction of tracer transport through computer simulation. State-of-the-art numerical algorithms of Lagrangian type simulate tracer transport by explicitly calculating the movement of individual particles within small regions of space. Tracer mass is conserved, meaning that no mass is artificially created or destroyed by the numerical calculations. This is a critical property for studies involving, e.g., contaminants, since even small concentrations can be toxic to humans, and any creation or degradation of the tracer must be due to physical and chemical processes and not to numerical artifacts. However, Lagrangian methods do not conserve the mass of the ambient fluid. This results in inaccurate tracer densities. That is, although tracer mass is conserved, its concentration is incorrectly computed, which can lead to serious inaccuracies in reaction dynamics and degradation in the predicted movement over time. The approach taken by the PI to resolve these difficulties is to consider the transport of both the tracer and ambient fluids, each of which must be conserved. The research is expected to result in significant improvement in the approximation of transport problems for long time simulation, and the training of at least one Ph.D. student in a multidisciplinary environment. This work has potential societal benefits as applied to problems in the contamination of ground-water, petroleum and natural gas production, and CO2 sequestration.