Analytical and Numerical Methods for Transport Equations

输运方程的分析和数值方法

• 批准号: 0104112
• 负责人: Guergana Petrova
• 金额: $ 8.17万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104112&HistoricalAwards=false
• 关键词: Analytical; Numerical; Methods; Transport; Equations

0104112PetrovaThis project addresses some problems in transport equations, Godunov type central schemes for hyperbolic conservation laws, classical and nonlinear approximation. The common thread that runs through the proposed research is the use of techniques from approximation theory, harmonic and functional analysis (Littlewood-Paley theory, wavelet decompositions, maximal functions, interpolation and K-functionals) to prove analytic results in other areas of applied mathematics and to use these techniques to develop numerical methods. Particular emphases will be placed on several issues: development of a satisfactory theory for linear transport equations in several space dimensions which arise when linearizing the nonlinear problem, development of Godunov type central schemes for solving multidimensional systems of conservation laws, and application of these schemes to various problems and models. Extensions of averaging lemmas to go from microscopic to macroscopic formulations will also be of primary concern. A portion of this project will address fundamental questions in nonlinear approximation and multivariate cubature.The areas under discussion (nonlinear approximation, analytical properties of solutions to transport equations and development of effective numerical methods for their computation) are of significant practical interest. The applications include image processing, statistical estimation, fluid mechanics, geophysics, meteorology, astrophysics, multi-component flows, ground water flow, semiconductors, and reactive flows.

本项目解决了输运方程、双曲守恒定律的Godunov型中心格式、经典近似和非线性近似中的一些问题。贯穿拟议研究的共同主线是使用来自近似理论、谐波和泛函分析（Littlewood-Paley理论、小波分解、极大函数、插值和k泛函）的技术来证明应用数学其他领域的分析结果，并使用这些技术来开发数值方法。特别的重点将放在几个问题上：在线性化非线性问题时出现的几个空间维度的线性输运方程的令人满意的理论的发展，用于解决守恒定律的多维系统的Godunov型中心格式的发展，以及这些格式在各种问题和模型中的应用。将平均引理从微观公式扩展到宏观公式也将是主要关注的问题。这个项目的一部分将解决非线性近似和多元文化中的基本问题。讨论的领域（非线性近似、输运方程解的解析性质和有效的数值计算方法的发展）具有重要的实际意义。应用包括图像处理、统计估计、流体力学、地球物理学、气象学、天体物理学、多组分流、地下水流、半导体和反应流。