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.

该项目研究守恒定律系统的二维黎曼问题，包括气体动力学的欧拉方程。 多维守恒定律理论还处于起步阶段，对黎曼问题的分析是表明多维系统中出现的奇点类型的第一步。 最近的数值模拟提供了马赫干形成点出乎意料的奇异行为的证据。 这些研究需要分析确认，并且可以从简单的原型示例中获得见解。 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（“亚音速区域”）。 亚音速解的分析产生了具有混合边界条件（狄利克雷和斜导数）的混合型偏微分方程中的自由边界问题。 早期的工作在冲击反射点附近局部解决了此类自由边界问题，但仅适用于一种简单的相互作用（正则反射）并且仅适用于简化系统（其中椭圆方程与双曲系统解耦）。 当前的项目将开发新技术来解决当边界条件不均匀倾斜以及亚音速系统中的方程不解耦时出现的更复杂的问题。 最后，对发散稀疏问题原型示例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。 更好的理解会对现实世界的应用产生影响。 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是基于守恒定律形式的偏微分方程。 理论进步，特别是在奇异行为分析方面的进步，将找到使数字代码更高效、更可靠的方法。 在开展这项研究的过程中，研究人员将成功地向新手研究人员（包括数学领域代表性不足的群体的成员）介绍守恒定律理论，并扩大他们的职业视野。