Analysis and Numerical Algorithms for Transport Equations and Related Problems

输运方程及相关问题的分析和数值算法

• 批准号: 0505501
• 负责人: Guergana Petrova
• 金额: $ 7.43万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505501&HistoricalAwards=false
• 关键词: Analysis; Numerical; Algorithms; Transport; Equations

Nonlinear transport equations model a variety of physical phenomenawhich often occur in real life. Examples of such phenomena arefluid mechanics, meteorology, reactive flows, multi-component flows, image processing, financial and biological modeling and others.Computing the solutions to these equations remains an important and challenging problem that requires development of fast, reliable and accurate numerical methods. At the same time, understanding the analytical properties of their solutions is not only useful for developing fast solvers, but remains imperative for real worldapplications. On the analytic side, we propose to utilize techniques from approximation theory and Harmonic Analysis to prove analytical results for nonlinear equations and to use these techniques to develop numerical methods.The techniques to be employed include Littlewood-Paley theory, wavelet decompositions, maximal functions, interpolation and K-functionals. Emphasis will be placed on the understanding the relation between micro- and macroscopic models for transport, where a main vehicle in moving from the micro to the macro level are averaging lemmas.On the numerical side, the project aims to further develop the Godunov type central schemes for multidimensional systems of conservation laws and related problems. These schemes do not employ Riemann problem solvers and characteristic decomposition. Their high resolution and simplicity turns them into a universal tool for solving a wide range of problems. The application of these schemes to multi-phase and multi-fluid flow models, the Saint-Venant systems of shallow water equations, multi-layer shallow water systems, shallow water equations on a rotatingsphere, granular material flows is another goal of the proposed research, which requires further development of central schemes and incorporation of various adaptive techniques into the central framework.

非线性传输方程模拟了各种物理现象经常在现实生活中发生。这种现象的示例是流体力学，气象，反应性流动，多组分流，图像处理，财务和生物建模等。为这些方程式提供解决方案仍然是一个重要且具有挑战性的问题，需要快速，可靠，可靠和准确的数值方法发展。同时，了解其解决方案的分析属性不仅对开发快速求解器有用，而且对现实世界申请仍然至关重要。在分析方面，我们建议利用近似理论和谐波分析的技术来证明非线性方程的分析结果，并使用这些技术来开发数值方法。要采用的技术包括Littlewood-Paley理论，小波，小波分解，最大功能，插入，插入，插入和K-functionals。将重点放在微观和宏观传输模型之间的关系上，在该模型中，从微观到宏观水平的主要工具是平均理血化的。在数值方面，该项目旨在进一步开发Godunov类型的中央方案，用于保存法律和相关问题的多维系统。这些方案不采用Riemann问题解决者和特征分解。他们的高分辨率和简单性将它们变成了解决广泛问题的通用工具。这些方案在多相和多流体流程模型中的应用，浅水方程的圣化系统，多层浅水系统，旋转旋转型的浅水方程，颗粒状材料流是拟议研究的另一个目标，这是需要进一步开发中央方案和各种适应性中心技术的进一步开发的研究。