Numerical Methods for Balance Laws with Applications to Shallow Water and Multiphase Flows

这个项目涉及平衡定律数值方法的发展、分析和实施。重点是对多维欧拉流体动力学方程进行平均而产生的系统。平均过程产生的非守恒项本质上是非守恒形式的，它的存在对理论和解的计算都有重大影响。计算上的困难包括解的非唯一性、严格双曲性的损失、离散非保守产品缺乏指导原则以及难以精确计算稳态解。本项目的目的是从数值的角度来理解这些不同的特征是如何相互联系的，以确定需要尊重的数值特征，并提供设计原则，从而产生简单有效的数值方案。该研究计划将通过研究一系列非保守模型来进行，并提供数值框架，在这些框架内可以解决计算困难：（i）解决双曲性损失和本征结构不可达性的松弛方法;（ii）离散平均模型，对封闭项给出精确估计，并为非保守产品提供明确的近似;（iii）用于两相流的混合算法，其在系数的不连续性上保持正确的跳跃条件。作为应用，本文方法将被用于研究两层水库间的交换流，并推广到多层浅水通过河道收缩流。分析预测是可用的，并将用于验证。本项目中考虑的类型的物理方程用于模拟各种流体流动。它们被用来模拟大规模的大气和海洋流动的地形，并描述地面流动的多孔介质。他们模拟河流和海岸流，大坝破裂和洪水，例如连接墨西哥湾和新奥尔良附近的庞恰特雷恩湖的里戈莱茨海峡的水流。它们用于水力学，描述和控制水库之间的交换流量。在多相流中，他们模拟液滴悬浮液的动力学，例如环境应用中的喷雾，或云动力学，这在可见性军事应用中可能很重要。除了实验和理论，数值模拟构成了这些应用领域的主要研究工具，本项目中解决的问题是其发展的基础。迄今为止，平衡定律的数值方法仍然面临着重大挑战，并提供不完整和不令人满意的答案。本项目中概述的方向解决了这些挑战，它们是原创的，它们是新的，它们强调可以导出到类似结构的其他系统的一般设计原则。拟议项目的完成预计将对这些社区的计算实践产生重大影响。

This project is concerned with the development, analysis and implementation of numerical methods for balance laws. The focus ison systems which arise by averaging the multidimensional Eulerequations of fluid dynamics. The averaging process producessource terms that are inherently in nonconservation form, whose presence has major consequences for the theory as well as forcomputations of solutions. Computational difficulties include nonuniqueness of solutions, loss of strict hyperbolicity, lack of guiding principles in discretizing nonconservative products and difficulties to compute accurately near steady-state solutions. The aim of this project is to understand, from a numerical view point, how these various characteristics are interconnected, to identify numerical features that need to be respectedand provide design principles which lead to simple and efficientnumerical schemes. This research program will be carried out by studyinga series of nonconservative models, and offer numerical frameworks within which the computational difficulties may be addressed: (i) a relaxation approach addressing the loss of hyperbolicity, and inaccessibility of eigenstructure; (ii) discrete averaging models, giving accurate estimates on closure terms and providing unambiguous approximations for nonconservative products; (iii) hybrid algorithms for two phase flows, which preserves the correct jump conditions across discontinuities in coefficients. As an application, the methods will be used to study two layer exchange flow between reservoirs, and extendedto multilayer shallow water flows through a channel contraction. Analytical predictions are available and will be used for validation.Physical equations of the type considered in this project are used to model a wide range of fluid flows. They are used to model large scale atmospheric and oceanic flows over terrains, and to describe ground flows in porous media. They model river and coastal flows, dam breaks and flooding, such as the flow in the Rigolets strait connecting the Gulf of Mexico and Lake Ponchartrain near New Orleans. They are used in hydraulics, to describe and control exchange flow between reservoirs. In multiphase flows, they model the dynamics of droplet suspensions such as sprays in environmental applications, or cloud dynamics, which may be important in visibility military applications. Alongside experimentsand theory, numerical simulations constitute a major tool of study in these applications areas, and the issues addressed in this project are fundamental to their development. To date, numerical methods for balance laws are still facing major challenges, and provide incomplete and unsatisfactory answers. The directions outlined in this project address these challenges, they are original, they are new and they emphasize general design principles that can be exported to other systems of similar structure. The completion of the proposed project is expected to have a significant impact on computational practices in these communities.