Development of Robust, Efficient and Highly Accurate Numerical Methods Based on Godunov-Type Central Schemes

基于Godunov型中心方案的鲁棒、高效和高精度数值方法的开发

• 批准号: 0610430
• 负责人: Alexander Kurganov
• 金额: $ 21.42万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0610430&HistoricalAwards=false
• 关键词: Development; Robust; Efficient; Highly; Accurate

Development of modern technology requires robust, efficient and highlyaccurate numerical methods for solving time-dependent partialdifferential equations, including multidimensional systems of hyperbolicconservation laws, balance laws, convection-reaction-diffusion equationsand related problems. A family of simple, universal and high-resolutionfinite-volume central schemes has been recently offered as an appealingalternative to more complicated and problem oriented upwind methods.The main goal of the project is applying the central schemes to variousmulti-phase and multi-fluid flow models, the Saint-Venant system ofshallow water equations (which describes flows in rivers and coastalareas), multi-layer shallow water equations (arising in oceanology),models of transport of pollutant in shallow water, several chemotaxismodels, reactive flows (in particular, the models describing stiffdetonation waves), shallow water equations on a rotating sphere,heterogeneous elasticity, granular material flows, dusty gas models(which describe volcanic eruptions), and others. Naturally, theseapplications, especially in the cases of high space dimensions, complexgeometries and moving boundaries/interfaces, require development andimplementation of additional numerical techniques such as differentadaption strategies, hybridization with Lagrangian-type methods,accurate and efficient operator splitting, numerical balancing betweenthe terms that are balanced in the original system of partialdifferential equations (development of well-balanced schemes), andothers that will be in the focus of the proposed research project.Central schemes have proved to be a reliable and robust tool for solvingmultidimensional systems of partial differential equations that describea variety of fundamental conservation laws in fluid mechanics, gasdynamics, geophysics, meteorology, magnetohydrodynamics, astrophysics,multi-component flows, granular flows, reactive flows, semiconductors,non-Newtonian flows, geometric optics, traffic flow, image processing,financial and biological modeling, differential games, optimal control,and many other areas. However, the models used in most practicalapplications are more complicated than just hyperbolic systems ofconservation laws, and therefore central schemes may only serve as abasis in designing robust, efficient and highly accurate numericalmethods. This project is aimed at developing a series of supplementarytechniques that are essential for the extension of applicability ofcentral schemes to many practically important problems, some of themare currently out of reach because the existing numerical methods areeither too inefficient/inaccurate or not applicable at all.

现代技术的开发需要强大，高效且高度准确的数值方法，以解决时间依赖时间的偏差方程，包括多维超固定法的多维系统，平衡法律，对流反应扩散方程和相关问题。最近已经提供了一个简单，普遍和高分辨率 - 体积的中央计划，以吸引更复杂和面向问题的上风方法。 （在海洋学出现），浅水中污染物的运输模型，几种趋化抗管模型，反应性流动（尤其是描述了僵硬波的模型），旋转球体上的浅水方程，异质弹性，颗粒状物质流动，尘土飞扬的气模型（描述了火山喷发）和其他。自然地，这些倾向，尤其是在高空间尺寸，复杂的界限和移动界限/界面的情况下，需要开发并进行其他数值技术（例如不同的趋于策略），以及与Lagrangian-type方法的混合，准确，高效的操作员分裂，均衡的均衡的均衡（均一型均衡）的原始术语（均具有均匀的均衡状态）均衡的计划），将成为拟议的研究项目的重点。事实证明，中央方案被证明是一种可靠且可靠的工具反应性流动，半导体，非牛顿流，几何光学，交通流量，图像处理，财务和生物建模，差异游戏，最佳控制以及许多其他领域。但是，在大多数实用应用中使用的模型比仅仅是保护定律的双曲线系统更为复杂，因此中央方案只能在设计强大，高效且高度准确的数值方法中充当憎恶。该项目旨在开发一系列的补充技术，这些补充技术对于将适用性的式中心方案扩展到许多实际重要的问题至关重要，目前有些实际上是无法触及的，因为现有的数值方法太效率低/不准确或完全不适用。