RUI: Multidimensional Conservation Laws and Related Applications

本项目由两部分组成。第一个项目旨在理解由多维守恒定律引起的混合（双曲-椭圆）型偏微分方程系统的解结构。用自相似坐标书写的多维守恒定律的一个显著特征是它们改变了类型：它们在远离原点的地方是双曲的，在靠近原点的地方是混合的。后者给我们提出了混合型非线性方程和自由边界的重要问题。特别是当波弱到非线性声波主导非线性熵波和涡量波时，激波极性分析无法解释激波反射的本质。这就是所谓的冯·诺伊曼悖论。这个项目的主要动机之一是解决渐近和计算分析的失败，以解决有关多维问题解的存在性和稳定性的某些悖论。理解多维守恒定律的数学结构是改进计算方法和解决这类悖论的关键一步。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，在其中提出非线性守恒定律，并开发有效的数值方法。第二个项目将研究各种稀疏数据野火蔓延模型的可行性，并开发有效的算法来解决模型问题。该项目将与加州里弗赛德的美国农业部森林火灾实验室进行沟通。野火传播模型继承了当地大气动力学（1公里及以上）和当地燃烧动力学（1米及以下）的尺度分离。此外，可用的数据是不规则的，有时是不准确的。现有的广泛模型需要完整的数据，计算量大，因此可能无法提供即时的结果，而这对有效的消防计划至关重要。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，从而可以立即和尽可能准确地规划灭火。多维守恒定律是物理和工程中基本过程的数学模型，如高速流动和超音速射流。对多维守恒定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等应用提供高效有效的方法。我们的野火模型将有助于有效的消防规划，从而为社会带来直接利益。该项目将在一所西班牙裔服务机构进行，并让本科生/硕士生参与模型问题的模拟，从而为学生在算法的设计、实现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（作为博士生）或他们未来在大洛杉矶地区高科技行业的工作做好准备。

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.