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

作者：suncica canic@gauss.Stanford.EDU注意日期：5/11/96 9：39 PM优先级：正常到：jjenkins at nsf11主题：如果您有任何意见，请告诉我，并希望我修改第二段。爱荷华州立大学代表理查德·哈斯布鲁克于周五签署了修订后的预算，每三年为2050万美元。下周一我会把表格传真给你。祝你周末愉快。爱荷华州立大学的Suncica Canic，Ames该项目涉及一维和二维空间中的非线性波相互作用。主要集中在跨音速流动中产生的波相互作用、楔形激波的反射、在多孔介质中的多相流以及化学反应流体。了解二维空间中非线性波相互作用的一个基准问题是激波反射问题。根据冲击强度和楔形角的不同，可能会出现不同的反射图案。悬而未决的问题涉及不同类型反射之间的转换标准，描述二维基本波的分叉图，以及提供描述可能奇点的解空间的存在理论。作为理解这些问题的第一步，本研究项目提出了两个空间维度中的自相似波相互作用的研究。该研究基于跨音速激波自由边界问题的非线性分析、双曲波相互作用中的准一维黎曼问题的波动曲线分析以及相应的波结构的数值模拟。该项目的结果将为理解跨音速激波的非线性稳定性和发展二维波相互作用的一般理论提供基本的贡献。在一维守恒律系统中，弱解的整体存在唯一性仍然是一个悬而未决的问题。弱溶液对扩散精确形式的精细依赖在区分物理相关的溶液(在油藏建模、化学反应流体、弹塑性变形和数值模拟中)方面起着至关重要的作用。在这个项目中，提出了一种有组织的方法来研究粘性的精确形式对弱解的存在和唯一性的影响。初步结果令人惊讶地指出了一些抛物型守恒律的对称破缺解。本项目所研究的数学问题来源于石油开采中的高性能计算模拟和跨音速流动的稳定性等应用。在这两种应用中，(非线性)波都是自然产生的(例如，油藏模拟中水和油之间界面的传播，超音速飞行中机翼周围冲击波的形成和传播，收敛-发散喷管中波的形成和相互作用)。用于计算机模拟这些现象的数值方法对某些参数很敏感。例如，产生人工(非物理)扩散的数值方案可能会产生误导性的答案，从而可能导致高生产成本。因此，了解解对问题中参数的依赖性(称为解的稳定性)对于设计有效的数值方法以产生物理上有意义的解至关重要。这个项目的重点是在联邦战略利益的许多领域中使用的一大类模型中出现的非线性波的稳定性。上面描述了两个这样的应用。由于问题的复杂性，需要使用各种最先进的理论技术(如自由边界问题的非线性分析，动力系统方法来研究扩散对弱黎曼解的影响)，并结合计算机模拟来解决这一问题。初步结果表明了以前从未研究过的新现象(例如，在跨音速流动中出现奇点和在采油中出现对称破缺解)。这些结果对于定制计算机代码以捕捉(“正确的”)物理行为至关重要。

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.