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.

小行星9401003 这个建议的动机是计算流体动力学中出现的数值困难，更普遍的是在连续介质力学中，当计算解决方案的非线性双曲方程组的PDE的。这样一个系统的解是不连续的，并且不是由它们的初始数据唯一确定的，通过所谓的熵准则来选择。 虽然熵准则在文献中被视为一个额外的条件，事后检查，P. I。开发的方法，其中熵准则在分析方案的性质或方案本身的推导中起着非常重要的作用。本文的重点是建立多维守恒律方程组的高精度格式的熵准则和非线性稳定性。 这项工作的目的是澄清的作用，所谓的熵准则的数值逼近双曲型系统的守恒律。私家侦探发展了新的分析方法和数值分析方法，建立了几种符合熵准则的数值格式。这样的结果保证了一个方法是非线性稳定的，并产生物理上有意义的间断解。该技术还提供了新的见解用于冲击波计算的计划，并建议可能的改进。这里的重点将是发展通量分裂方法，二阶Godunov型计划，和真正的多维计划的守恒律系统。分析将包括的概念，如：熵耗散估计，跟踪单调区域，补偿紧致方法，测量值的解决方案，等进一步了解的非线性稳定性将通过考虑的问题的唯一性和连续依赖的初始条件。