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.

该项目研究了保护法系统的二维Riemann问题，包括气体动力学的Euler方程。 多维保护定律的理论仍处于起步阶段，对黎曼问题的分析是第一步，它将指示多维系统中出现的奇异性类型。 最近的数值模拟提供了在马赫茎的形成点出乎意料的奇异行为的证据。 这些研究要求进行分析确认，并将从简单的原型示例中获得洞察力。 在自相似方法中，系统会改变类型 - 双曲线远离原点（“超音速流”）和附近的混合双曲线 - 涡流类型（“亚音速区域”）。 对亚音速溶液的分析产生了具有混合边界条件（dirichlet和倾斜衍生物）的混合型部分微分方程中的自由边界问题。 较早的工作解决了在冲击反射点附近的本地自由边界问题，但仅用于一种简单的相互作用（常规反射），仅用于简化的系统（椭圆方程与双曲线系统解耦）。 当前的项目将开发新技术，以解决边界条件不统一的更复杂的问题，以及亚音速系统中的方程式不脱离时。 最后，研究稀疏问题的原型示例将揭示出奇异的马赫茎形成点。 保护法模型模型空气动力学和连续力学中的基本问题。 更好的理解在现实世界的应用中会产生后果。 例如，构成天气和气候预测和核反应堆安全分析基础的大规模数值模拟是基于保护法律形式的部分微分方程。 理论进步，特别是在分析奇异行为时，将找到使数值代码更有效和更可靠的方式。 在进行这项研究时，研究人员将在介绍初学者的研究人员中取得成功，其中包括数学中代表性不足的群体成员，将其介绍给保护法理论，并扩大其职业视野。