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。这些方法保留了局部示踪剂的质量，而不是周围流体的质量。因此，他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和同事最近为最简单的输运问题定义了体积校正特性混合方法（VCCMM），通过考虑输运问题而不是将其视为示踪剂的单个双曲线方程，而是将其视为描述示踪剂和环境流体运动的两个方程的系统，每个方程都必须在局部守恒。本项目将：(1)完成VCCMM的开发，将其建立在良好的理论基础上，并开发软件的并行版本；(2)改进并扩展VCCMM到更复杂的流程；(3)开发网格适配版；(4)发展非线性输运问题的方法，包括混相、可压缩流动和两相、非混相流动，这些问题的解决方案可能包含激波和稀疏。由于离散近似具有较少的数值扩散，并保留了重要的物理原理，预计该项目将对长时间模拟输运问题的近似有重大改进。该项目预计将产生更广泛的影响，包括：(1)开发一种科学的软件工具，可以应用于广泛的实际问题；(2)在多学科环境下培养1名博士生；(3)社会效益，可以更好地模拟地质盆地形成、长寿命的放射性同位素衰变、混相指化和两相流。预测一种化学物质（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，需要在地下水污染物迁移研究中出现。本课题通过计算机模拟研究如何改进对示踪剂输运的预测。最先进的拉格朗日型数值算法通过明确计算空间小区域内单个粒子的运动来模拟示踪剂输运。示踪剂的质量是守恒的，这意味着数值计算不会人为地创造或破坏任何质量。这是涉及污染物等研究的一个关键特性，因为即使是很小的浓度也可能对人类有毒，而且示踪剂的任何产生或降解都必须是由于物理和化学过程，而不是由于数值伪影。然而，拉格朗日方法不能保存周围流体的质量。这导致了不准确的示踪剂密度。也就是说，虽然示踪剂的质量是守恒的，但它的浓度计算是错误的，这可能导致反应动力学的严重不准确，并随着时间的推移导致预测运动的退化。PI为解决这些困难所采取的方法是同时考虑示踪剂和环境流体的输送，每一种都必须保持守恒。该研究有望在长时间模拟中近似运输问题方面取得重大进展，并在多学科环境中培养至少一名博士生。这项工作对于解决地下水污染、石油和天然气生产以及二氧化碳封存等问题具有潜在的社会效益。

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.