Collaborative Research: Numerical methods for Shallow Water Equations and Related Models

合作研究：浅水方程及相关模型的数值方法

• 批准号: 1216957
• 负责人: Alexander Kurganov
• 金额: $ 20万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216957&HistoricalAwards=false
• 关键词: Collaborative; Research; Numerical; methods; Shallow

The project is aimed at developing accurate, efficient, and robust numerical methods for shallow water equations and related models, with particular reference to problems that admit non-smooth (discontinuous) solutions and to problems that involve highly disparate scales and therefore are difficult to solve numerically. Shallow water and related models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. These models are systems of time-dependent partial differential equations (PDEs) that are derived using physical properties such as conservation of mass and momentum, and hydrostatic or barotropic approximations. The principal part of the proposed research will be focused on the development of new methods for solving problems involving complicated nonlinear wave phenomena, problems with complex computational domains and moving interfaces. The resulting methods, while based on high-order shock-capturing finite-volume schemes and non-dissipative mesh-free particle methods, will incorporate special numerical techniques such as numerical balancing between the terms that are balanced in the original system of PDEs (development of well-balanced schemes), ensuring positivity of all fluid layers (this is absolutely necessary for both accurate description of dry and near dry states and enforcement of nonlinear stability), accurate and efficient operator splitting, accurate and efficient interface tracking, and others that will be in the focus of the proposed research project. The proposed project will contribute significantly toward development of computational methods for shallow water and related models. Special attention will be paid to applications arising in oceanography and atmospheric sciences, in which the Coriolis forces (due to the Earth's rotation), thermodynamics, and turbulent effects have to be taken into account. The problems under study include, among others, formation and propagation of atmospheric fronts and ocean currents, propagation of tsunami waves and their on-shore arrival, as well as propagation of pollutants in various environments. The numerical methods under design will provide considerably more powerful tools for studying a variety of internal and surface water waves, including tsunami and rogue waves. These extreme waves, which arise both in deep and shallow water, have a significant impact on the safety of people and infrastructure, and are responsible for damage to ships, oil platforms, coastlines, and sea bottoms and for changes to the biological environment. Thus, understanding the physics of these extreme waves is an important task that may even contribute to saving lives.

该项目旨在为浅水方程和相关模型开发准确，高效且鲁棒的数值方法，特别是提及接受非平滑（不连续）解决方案的问题，以及涉及高度降低量表的问题数值。浅水和相关模型被广泛用作数学框架，以研究河流和沿海地区的水流，并研究大气科学和海洋学中的各种现象。这些模型是使用物理特性（例如质量和动量保护）以及静液压或压缩近似值等物理性能得出的时间依赖性部分微分方程（PDE）的系统。拟议研究的主要部分将集中于解决涉及复杂非线性波现象，复杂计算域和移动界面的问题的新方法的发展。最终的方法基于高阶捕捉有限体积方案和非缺失的无网格粒子方法，将结合特殊的数值技术，例如在PDES原始系统中平衡的术语之间的数值平衡（开发（开发）（开发）均衡的方案），确保所有流体层的积极性（这对于精确描述干燥和近干状态以及非线性稳定性的执行是绝对必要的）将是拟议的研究项目的重点。拟议的项目将有助于开发浅水和相关模型的计算方法。必须特别注意在海洋学和大气科学中产生的应用，必须考虑科里奥利力量（由于地球旋转），热力学和湍流效应。研究的问题包括大气前沿和洋流的形成和繁殖，海啸波的传播及其在近海到达以及在各种环境中污染物的繁殖。设计下的数值方法将为研究各种内部和地表水波（包括海啸和流氓波浪）提供更强大的工具。这些极端的波浪在深水和浅水中都产生，对人和基础设施的安全产生了重大影响，并负责损坏船舶，石油平台，海岸线和海底，以及对生物环境的变化。因此，了解这些极端波的物理学是一项重要的任务，甚至可能有助于挽救生命。