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New Tools in the Study of Wave Propagation: Dynamical Systems for Kinetic Equations, Inviscid Limits for Modulated Periodic Waves, and Rigorous Numerical Stability Analysis

波传播研究的新工具：运动方程的动力系统、调制周期波的无粘极限以及严格的数值稳定性分析

基本信息

批准号：
1700279
负责人：
Kevin Zumbrun
金额：
$ 20.8万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700279&HistoricalAwards=false
关键词：
New Tools Study Wave Propagation

项目摘要

The PI will study a selection of key open problems in the theory of wave propagation arising in a variety of settings including thin film flow, pattern formation, detonation, and the kinetic theory of gases. The problems considered share the features of computational complexity, multiple length/time scales, and genuine physical interest in applications. Many concern questions that current numerics and experiment are not adequate to resolve. Several of the planned subprojects involve numerically assisted proof using scientific computation with guaranteed error bounds, up to and including rigorous numerical proof. An integral part of the project is the simultaneous development of a user-friendly numerical platform, STABLAB, for numerical stability investigation, and the systematic exploration with this platform of physical behavior in gas and fluid dynamics in the delicate situations of reacting or ionized flow. The problems addressed involve issues in dynamical systems, singular perturbation theory, spectral theory of linear operators, nonlinear partial differential equations, and rigorous scientific computation, and should result in the development of new mathematical tools of general application. In particular, development of dynamical systems tools for kinetic shock and boundary layer problems would unify and extend results obtained for Boltzmann phenomena by the "Kyoto School" of Sone et al using a variety of formal and analytic methods. Likewise, the introduction of new inviscid stability criteria for roll waves and of Kreiss symmetrizer techniques for analysis of modulated fronts open new directions in the study of periodic modulation. The problem on galloping detonations, if solved, will answer a longstanding question, while associated rigorous Wensel,Kramers, and Brillouin method developments will be of wide general use. Determination of simple stability criteria for roll waves in shallow water flow are of practical interest in hydraulic engineering. Finally, the development of rigorous numerical proof and error estimate techniques is potentially transformative, having broader implications for standards in scientific computing.

PI 将研究在各种环境中出现的波传播理论中的一系列关键开放问题，包括薄膜流动、图案形成、爆炸和气体动力学理论。所考虑的问题具有计算复杂性、多个长度/时间尺度以及对应用的真正物理兴趣的特征。许多问题涉及当前的数值和实验不足以解决的问题。几个计划的子项目涉及使用科学计算进行数值辅助证明，并保证误差范围，直至并包括严格的数值证明。该项目的一个组成部分是同时开发一个用户友好的数值平台 STABLAB，用于数值稳定性研究，并利用该平台对反应或电离流的微妙情况下气体和流体动力学的物理行为进行系统探索。所解决的问题涉及动力系统、奇异摄动理论、线性算子谱理论、非线性偏微分方程和严格的科学计算等领域的问题，并应导致开发出通用的新数学工具。特别是，针对动冲击和边界层问题的动力系统工具的开发将统一和扩展 Sone 等人的“京都学派”使用各种形式和分析方法获得的玻尔兹曼现象的结果。同样，引入新的滚波无粘稳定性准则和用于分析调制前沿的 Kreiss 对称化技术，为周期性调制的研究开辟了新的方向。疾驰爆炸问题如果得到解决，将回答一个长期存在的问题，而相关的严格的文塞尔、克莱默和布里渊方法的发展将具有广泛的通用性。确定浅水流中滚波的简单稳定性准则对于水利工程具有实际意义。最后，严格的数值证明和误差估计技术的发展具有潜在的变革性，对科学计算标准具有更广泛的影响。