Analysis and Numerical Algorithms for Transport Equations and Related Problems

非线性输运方程模拟了真实的生活中经常发生的各种物理现象。这些现象的例子是流体力学，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的和具有挑战性的问题，需要开发快速，可靠和准确的数值方法。同时，了解其解的解析性质不仅对开发快速求解器有用，而且对于真实的世界应用仍然是必不可少的。在分析方面，我们建议利用近似理论和调和分析的技术来证明非线性方程的分析结果，并使用这些技术来发展数值方法。 所采用的技术包括Littlewood-Paley理论、小波分解、极大函数、插值和K-泛函。重点将放在理解微观和宏观运输模型之间的关系，其中一个主要的车辆在移动从微观到宏观水平的平均lemmas.On数值方面，该项目旨在进一步发展Goddom型的多维系统的守恒律和相关问题的中心计划。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。这些计划的应用程序的多相和多流体流动模型，圣维南系统的浅水方程，多层浅水系统，浅水方程上的一个球，颗粒物质流是拟议的研究的另一个目标，这需要进一步发展的中心计划和纳入到中心框架的各种自适应技术。

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.