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

[401003] Le Floch这个建议是由计算流体力学中，更普遍的是在连续介质力学中，当计算非线性双曲PDE系统的解时，出现的数值困难所激发的。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决定的，是通过所谓的熵准则来选择的。虽然熵标准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，其中熵标准在分析方案的性质或在方案本身的推导中起着非常重要的作用。本文的重点是建立具有守恒律的多维系统的熵准则和高阶精确格式的非线性稳定性。这项工作的目的是澄清所谓的熵准则的作用，为数值逼近双曲系统的守恒定律。他们开发了新的分析方法和数值分析方法，并确定了几种数值格式符合熵准则。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。这里的重点将是发展通量分裂方法，二阶godunov型格式，以及守恒律系统的真正多维格式。分析将包括：熵耗散估计、单调区域跟踪、补偿紧性方法、测度值解等概念。通过考虑关于初始条件的唯一性和连续依赖性问题，将进一步了解非线性稳定性。

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.