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

PI将研究在各种环境中产生的波传播理论中的关键开放问题，包括薄膜流，图案形成，爆震和气体动力学理论。 所考虑的问题共享的计算复杂性，多个长度/时间尺度，和真正的物理利益的应用程序的功能。 许多关注的问题，目前的数字和实验是不足以解决。 计划中的几个子项目涉及使用科学计算的数值辅助证明，保证误差范围，直到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，并在反应或电离流的微妙情况下系统地探索气体和流体动力学的物理行为。 所解决的问题涉及动力系统，奇异摄动理论，线性算子的谱理论，非线性偏微分方程和严格的科学计算的问题，并应导致新的数学工具的发展的一般应用。特别是，发展动力学系统的工具，为动力冲击和边界层问题将统一和扩展的结果得到玻尔兹曼现象的“京都学校”的Sone等人使用各种形式和分析方法。同样，新的无粘稳定性准则的推出，滚波和Kreiss对称化器技术分析调制前打开新的方向在研究周期性调制。驰爆问题，如果解决，将回答一个长期存在的问题，而相关的严格的文塞尔，克莱默斯和布里渊方法的发展将是广泛的普遍使用。确定浅水中滚波的简单稳定性准则在水利工程中具有重要的实际意义。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。

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.