Prototype Systems of Multidimensional Conservation Laws

多维守恒定律原型系统

• 批准号: 0807569
• 负责人: Barbara Keyfitz
• 金额: $ 17.5万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2009-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807569&HistoricalAwards=false
• 关键词: Prototype; Systems; Multidimensional; Conservation; Laws

This project studies two-dimensional Riemann problems for systems of conservation laws, including the Euler equations of gas dynamics. The theory of multi-dimensional conservation laws is in its infancy, and this analysis of Riemann problems is a first step that will indicate the types of singularities that arise in multi-dimensional systems. Recent numerical simulations have presented evidence of unexpectedly singular behavior at the formation points of Mach stems. These studies call for analytical confirmation, and insight will be gained from simple prototype examples. In the self-similar approach, the systems change type -- hyperbolic far from the origin ("supersonic flow") and of mixed hyperbolic-elliptic type near the origin ("subsonic region"). Analysis of the subsonic solution gives rise to free boundary problems in mixed-type partial differential equations with mixed boundary conditions (Dirichlet and oblique derivative). Earlier work solved such free boundary problems locally near shock reflection points, but only for one simple interaction (regular reflection) and only for simplified systems (where the elliptic equation decoupled from the hyperbolic system). The current project will develop new techniques for the more complicated problems that arise when the boundary condition is not uniformly oblique, and when the equations in the subsonic system do not decouple. Finally, study of a prototype example of diverging rarefactions problems will shed light on singular Mach stem formation points. Conservation laws model fundamental problems in aerodynamics and continuum mechanics. Better understanding has consequences in real world applications. For example, the large-scale numerical simulations that form the basis of weather and climate prediction and of nuclear reactor safety analysis are based on partial differential equations in the form of conservation laws. Theoretical advances, particularly in the analysis of singular behavior, will find their way into making numerical codes more efficient and more reliable. In carrying out this research, the investigators will build on success in introducing beginning researchers, including members of groups underrepresented in mathematics, to the theory of conservation laws, and in expanding their career horizons.

这个项目研究守恒律系统的二维黎曼问题，包括气体动力学的欧拉方程。 多维守恒律理论还处于起步阶段，对黎曼问题的分析是指出多维系统中出现的奇点类型的第一步。 最近的数值模拟已经提供了在马赫茎形成点处意外奇异行为的证据。 这些研究需要分析确认，并从简单的原型示例中获得洞察力。 在自相似方法中，系统改变类型--远离原点的双曲型（“超音速流”）和靠近原点的混合双曲椭圆型（“亚音速区”）。 亚音速解的分析引起混合型偏微分方程的自由边界问题与混合边界条件（狄利克雷和斜导数）。 早期的工作解决了这样的自由边界问题局部附近的激波反射点，但只有一个简单的相互作用（定期反射），并仅为简化系统（其中椭圆方程解耦的双曲系统）。 目前的项目将为边界条件不是均匀倾斜的和亚音速系统中的方程不解耦时出现的更复杂的问题开发新技术。 最后，发散稀疏问题的原型例子的研究将揭示奇异马赫干形成点。 守恒定律模拟了空气动力学和连续介质力学中的基本问题。 更好的理解在真实的世界应用中具有重要意义。 例如，构成天气和气候预测以及核反应堆安全分析基础的大规模数值模拟就是以守恒定律形式的偏微分方程为基础的。 理论的进步，特别是在奇异行为的分析，将找到自己的方式，使数字代码更有效，更可靠。 在进行这项研究时，研究人员将成功地介绍开始的研究人员，包括在数学中代表性不足的群体的成员，守恒定律的理论，并扩大他们的职业视野。