Mathematical Sciences: Hyperbolic Systems of Conservation Laws: Theory and Applications

数学科学：守恒定律的双曲系统：理论与应用

• 批准号: 9401003
• 负责人: Philippe Le Floch
• 金额: $ 6万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401003&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hyperbolic; Systems; Conservation

9401003 Le Floch This proposal is motivated by the numerical difficulties arising in Computational Fluid Dynamics, and more generally in continuum mechanics, when computing solutions to nonlinear hyperbolic systems of PDE's. The solutions to such a system are discontinuous, and, being not uniquely determined by their initial data, are selected via the so-called entropy criterion. While the entropy criterion is treated in the literature as an extra condition that is checked afterwards, the P.I. developed approaches where the entropy criterion plays the very central role in analyzing the properties of a scheme, or in the derivation of the scheme itself. The emphasis here will be on establishing the entropy criterion and the nonlinear stability of high order accurate schemes for multidimensional systems of conservation laws. This work is aimed at clarifying the role of the so-called entropy criterion for the numerical approximation of hyperbolic systems of conservation laws. The P.I. developed new methods of analysis and numerical analysis, and established that several numerical schemes are consistent with the entropy criterion. Such a result guarantees that a method is nonlinearly stable, and produces the physically meaningful discontinuous solutions. The techniques also provide new insight on the schemes used for shock wave calculations, and suggest possible improvements. The focus here will be on developing flux-splitting methods, second-order Godunov-type schemes, and truly multidimensional schemes for systems of conservation laws. The analysis will include concept such as: entropy dissipation estimates, tracking of monotonicity regions, the compensated compactness method, the measure-valued solutions, etc. Further insight about the nonlinear stability will be obtained by considering the issue of uniqueness and continuous dependence with respect to the initial condition.

9401003 Le Floch 该提议的动机是计算流体动力学中，更普遍的是连续介质力学中，在计算偏微分方程非线性双曲系统的解时出现的数值困难。这种系统的解是不连续的，并且不是由其初始数据唯一确定的，而是通过所谓的熵准则选择的。 虽然熵准则在文献中被视为事后检查的额外条件，但 P.I.开发了一些方法，其中熵准则在分析方案的属性或方案本身的推导中起着非常核心的作用。这里的重点是建立多维守恒定律系统的熵准则和高阶精确格式的非线性稳定性。 这项工作旨在阐明所谓的熵准则在守恒定律双曲系统数值逼近中的作用。 P.I.开发了新的分析和数值分析方法，并建立了几个与熵准则一致的数值格式。这样的结果保证了方法是非线性稳定的，并产生了物理上有意义的不连续解。这些技术还为冲击波计算方案提供了新的见解，并提出了可能的改进建议。这里的重点将是开发通量分裂方法、二阶戈杜诺夫型方案以及守恒定律系统的真正多维方案。分析将包括以下概念：熵耗散估计、单调性区域跟踪、补偿紧致性方法、测量值解等。通过考虑唯一性和连续依赖性问题，将获得关于非线性稳定性的进一步见解到初始条件。