Mathematical Sciences: Nonlinear Wave Interactions in One and Two Space Dimensions

作者：Suncica Canic canic@gauss.斯坦福大学.EDU在注日期： 1996年5月11日9：39 PM优先级：正常收件人：jjenkins at nsf 11主题： - 留言内容-亲爱的乔， 请按您要求的格式附上我的项目摘要。 让我知道如果你有任何意见，并希望我 修改第二段。 三年中每年2050美元的修订预算已经 爱荷华州州立大学代表理查德·哈斯布鲁克星期五签署了这项协议。 下周一我会把表格传真给你。 周末愉快。 此致， 松契察 项目摘要 “一维和二维空间中的非线性波相互作用” 爱荷华州州立大学，艾姆斯 本项目研究一维和二维空间中的非线性波相互作用 尺寸.主要的焦点是跨音速中产生的波的相互作用 流动，楔体对激波的反射，多孔介质中的多相流 介质和化学反应性流体。理解的基准问题 二维空间中非线性波的相互作用就是激波 反射问题。取决于冲击强度和楔角 可以出现不同的反射图案。关于过渡的公开问题 不同反射类型之间的准则，分叉图 描述二维基本波，以及提供描述可能奇点的解空间的存在理论。 该研究项目提出了在两个空间维度中研究自相似波相互作用，作为理解这些问题的第一步。 本文基于自由边界问题的非线性分析， 跨音速激波，关于准一维波的曲线分析 双曲波相互作用中出现的Riemann问题，以及 相应的波浪结构的数值模拟。的结果 他的项目将为理解 跨音速激波的非线性稳定性和一个 二维波相互作用的一般理论。 在一维守恒律系统中，整体存在性和 弱解的唯一性仍然是一个悬而未决的问题。微妙的依赖 弱解的精确形式的扩散起着至关重要的作用， 区分物理上相关的解决方案（在油藏建模中， 化学反应性流体，弹性塑性变形，以及数值模拟 模拟）。在这个项目中， 粘性的精确形式对解的存在唯一性的影响 提出了弱解。初步结果令人惊讶 指出了某些抛物型守恒律的对称破缺解。 在这个项目中研究的数学问题来自于应用 例如石油开采中的高性能计算模拟， 跨音速流动稳定性在这两种应用（非线性）波出现 自然地（例如，油水界面的扩展 在油藏模拟中，冲击波的形成和传播 超音速飞行中机翼周围的波的形成和相互作用 收缩-扩张喷嘴）。计算机中使用的数值方法 这些现象的模拟对某些参数敏感。为 例如，产生人工（非物理）扩散的数值方案 可能产生误导性的答案，这可能导致高的生产成本。 因此，理解解对参数的依赖性， 一个问题（称为解的稳定性）在设计有效的 能产生物理上有意义的解的数值方法 该项目的重点是非线性波的稳定性， 出现在联邦战略利益的许多领域中使用的大量模型中。上面描述了两个这样的应用。因为 问题的复杂性，各种最先进的理论 技术（例如，自由边界问题的非线性分析，动力学的 扩散对弱黎曼解影响的系统方法） 再加上计算机模拟，需要用来解决这个问题。 初步结果表明了以前没有研究过的新现象 (e.g.,跨音速流中奇点出现与对称性 油回收中的破碎溶液）。这些结果至关重要 裁剪计算机代码以捕捉（“正确”）物理行为。

Author: Suncica Canic canic@gauss.Stanford.EDU at NOTE Date: 5/11/96 9:39 PM Priority: Normal TO: jjenkins at nsf11 Subject: ------------------------------- Message Contents ------------------------------- Dear Joe, Please find attached the abstract of my project in the format you requested. Let me know if you have any comments and would like me to revise the second paragraph. The revised budget of $20.5K for each of three years has been signed on Friday by Richard Hasbrook, Iowa State University representative. I will fax you the forms this coming Monday. Have a nice weekend. Regards, Suncica ABSTRACT OF THE PROJECT "Nonlinear Wave Interactions in One and Two Space Dimensions" Suncica Canic, Iowa State University, Ames This project deals with nonlinear wave interactions in one and two space dimensions. The main focus is on wave interactions that arise in transonic flow, reflections of a shock by a wedge, multiphase flow through the porous media and chemically reactive fluids. A benchmark problem for understanding the interaction of nonlinear waves in two-space dimensions is the shock reflection problem. Depending on the shock strength and the wedge angle different reflection patterns can occur. Open problems regard the transition criteria between different types of reflection, bifurcation diagram describing two-dimensional elementary waves, and the existence theory that would provide a solution space describing possible singularities. This research project proposes a study of self-similar wave interactions in two space dimensions as a first step towards understanding these issues. The study is based on the nonlinear analysis of free-boundary problems for transonic shock waves, on the wave curve analysis of quasi-one-dimensional Riemann problems that arise in hyperbolic wave interactions, and on numerical simulations of the corresponding wave structures. The results from t his project would provide a fundamental contribution towards understanding the nonlinear stability of transonic shock waves and the development of a general theory of two-dimensional wave interactions. In one-dimensional systems of conservation laws, global existence and uniqueness of weak solutions is still an open question. A delicate dependence of weak solutions on the precise form of diffusion plays a crucial role in distinguishing physically relevant solutions (in oil reservoir modeling, chemically reactive fluids, elastic plastic deformation, and in numerical simulations). In this project an organized approach to the study of the influence of the precise form of viscosity on the existence and uniqueness of weak solutions has been proposed. Preliminary results surprisingly indicate symmetry breaking solutions for some parabolic conservation laws. Mathematical issues studied in this project derive from applications such as high-performance computing simulations in oil recovery, and the stability of transonic flow. In both applications (nonlinear) waves arise naturally (e.g., propagation of an interface between water and oil in oil reservoir simulations, formation and propagation of shock waves around a wing in a supersonic flight, formation and interaction of waves in a converging-diverging nozzle). Numerical methods used in computer simulations of these phenomena are sensitive to certain parameters. For example, numerical schemes that produce artificial (non-physical) diffusion may produce misleading answers that may result in high production costs. Therefore, understanding the dependence of solutions on the parameters in a problem (called the stability of solutions) is crucial in devising efficient numerical methods that would produce physically meaningful solutions. This project focuses of the stability of nonlinear waves that arise in a large class of models used in many areas of Federal strategic interests. Two such appl ications are described above. Because of the complexity of the problem, various state-of-the-art theoretical techniques (e.g., nonlinear analysis of free-boundary problems, dynamical systems approach to the influence of diffusion on weak Riemann solutions) coupled with computer simulations, need to be used to tackle the problem. Preliminary results indicate new phenomena that have not been studied before (e.g., the occurrence of singularities in transonic flow and symmetry breaking solutions in oil recovery). These results are crucial in tailoring computer codes to capture the ("correct") physical behavior.