RUI: Multidimensional Conservation Laws and Related Applications

本项目由两部分组成。第一个项目旨在了解多维守恒律引起的偏微分方程的混合（双曲-椭圆）型系统的解结构。在自相似坐标系中的多维守恒律的一个显著特征是它们改变了它们的类型：它们在远离原点的地方是双曲的，而在原点附近是混合的。后者使我们面临混合型非线性方程和自由边界的重要问题。特别是，当波足够弱，非线性声波占主导地位的非线性熵和涡波，激波极分析无法解释的激波反射的性质。这就是所谓的冯·诺依曼悖论。 这是这个项目的主要动机之一，以解决渐近和计算分析的失败，以解决某些悖论的存在和稳定的多维问题的解决方案。理解多维守恒律的数学结构是改进计算方法和解决这些悖论的关键一步。该项目旨在研究这些混合型问题，以获得新的物理见解，开发新的分析工具，并找到正确的数学框架，提出非线性守恒律，并开发有效的数值方法。第二个项目将调查各种野火蔓延模型的可行性与稀疏数据，并开发有效的算法来解决模型问题。该项目将与位于加利福尼亚州滨江的美国农业部森林火灾实验室进行沟通。 野火蔓延模型继承了当地大气动力学（一公里以上）和当地燃烧动力学（一米以下）的尺度分离。此外，现有数据不规则，有时不准确。现有的广泛的模型，需要完整的数据，是计算密集型的，因此可能无法提供即时的结果，这是至关重要的有效的消防计划。 该项目将为简化模型开发有效的算法，这些算法可以与稀疏数据相结合，以便能够立即并尽可能准确地规划消防工作。多维守恒定律是物理和工程中基本过程的数学模型，例如高速流动和超音速射流。对多维守恒律的深入理解将为可压缩气体动力学、热力学、多相流和多孔介质流动等应用提供高效、有效的方法。我们的野火建模将有助于有效的消防规划，从而为社会带来直接利益。该项目将在西班牙裔服务机构进行，并涉及本科生/硕士生的模型问题的模拟，从而为学生在算法的设计，实施和开发方面的进一步工作做好准备。该项目将为学生提供培训，为他们的学术生涯做好准备（作为博士）。学生），或他们未来在大洛杉矶地区高科技行业的工作。

This project consists of two parts. The first project aims at understanding the solution structures of mixed (hyperbolic-elliptic) type systems of partial differential equations arising from multidimensional conservation laws. A distinctive feature of multidimensional conservation laws written in self-similar coordinates is that they change their type: they are hyperbolic far from the origin, and mixed near the origin. The latter confronts us with important problems in nonlinear equations of mixed types and free boundaries. In particular, when the waves are weak enough that the nonlinear acoustic waves dominate the nonlinear entropy and vorticity waves, shock polar analysis fails to explain the nature of shock reflection. This is the so called von Neumann paradox. It is one of the main motivations of this project to address the failures of asymptotic and computational analysis to resolve certain paradoxes concerning the existence and the stability of solutions of multidimensional problems. Understanding the mathematical structure of multidimensional conservation laws is a crucial step in improving computational methods and in resolving such paradoxes. The project is directed at investigating these mixed type problems to gain new physical insights, to develop novel analytical tools, and to find the correct mathematical framework in which to pose the nonlinear conservation laws and to develop efficient numerical methods. The second project will investigate the feasibility of various wildfire spread models with sparse data, and development of efficient algorithms to solve the model problems. This project will be conducted in communication with the USDA Forest Fire Lab in Riverside, CA. The wildfire spread models inherit scale separation of the local atmospheric dynamics (one kilometer and larger) and the local combustion dynamics (one meter and smaller). Also, available data are irregular and sometimes inaccurate. Existing extensive models, require complete data, are computationally intensive, and thus may not provide immediate results, which are crucial for effective fire-fighting plans. The project will develop efficient algorithms for simplified models, which can be incorporated with sparse data, so that the fire-fighting could be planned immediately and as accurately as possible.Multidimensional conservation laws are mathematical models for fundamental processes in physics and engineering, such as high-speed flows and supersonic jets. A deeper understanding of multidimensional conservation laws will provide efficient and effective methods for applications, such as compressible gas dynamics, thermodynamics, multi-phase flow and porous medium flow. Our wildfire modeling will contribute to effective fire-fighting planning and thus will have direct benefits for society. This project will take place at a Hispanic-serving institution, and involve undergraduate/master students in simulations of the model problems, thus preparing the students for further work in the design, implementation, and development of algorithms. This project will provide students with training to prepare them for their academic careers (as Ph.D. students), or their future jobs in the high tech industry in the greater Los Angeles area.