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

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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.