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.

非线性输运方程模拟了现实生活中经常发生的各种物理现象。此类现象的例子包括流体力学、气象学、反应流、多组分流、图像处理、金融和生物建模等。计算这些方程的解仍然是一个重要且具有挑战性的问题，需要开发快速、可靠和准确的数值方法。同时，了解其解决方案的分析属性不仅对于开发快速求解器有用，而且对于现实世界的应用仍然至关重要。在解析方面，我们建议利用近似理论和调和分析技术来证明非线性方程的解析结果，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、小波分解、极大函数、插值和 K 泛函。重点将放在理解运输的微观和宏观模型之间的关系，其中从微观到宏观的主要车辆是平均引理。在数值方面，该项目旨在进一步开发戈杜诺夫型中心守恒定律多维系统的方案及相关问题。这些方案不采用黎曼问题求解器和特征分解。它们的高分辨率和简单性使它们成为解决各种问题的通用工具。这些方案在多相和多流体流动模型、浅水方程的圣维南系统、多层浅水系统、旋转球上的浅水方程、颗粒物质流中的应用是该研究的另一个目标，这需要进一步开发中心方案并将各种自适应技术纳入中心框架。