Abstract DMS-9704957 PI: Shi Jin Title: Numerical Methods for Hyperbolic Systems and Related Problems I propose to study and design numerical methods for hyperbolic systems of conservation laws and related problems. The underlying physical problems arise from fluid dynamics, rarefied gas dynamics and wave propagation. I plan to work on three main projects: 1. the development of robust shock capturing methods for hyperbolic systems with relaxations, including kinetic equations; 2. the construction and study of relaxation approximations and relaxation schemes for viscous systems of conservations laws and Hamilton-Jacobi equations. 3. asymptotic and numerical study on wave propagation in random media. These projects if successfully carried out will provide attractive numerical methods that are able to simulate a wide variety of challenging physical and industrial problems. They also help to gain more understanding of the underlying physical phenomena. With the development of modern computers, scientific computation has been playing a central role in scientific investigation. The physical and industrial problems we study here arise in complex fluids, optics and materials, which can not be solved by other scientific tools. Nervetheless, numerical computation has been demonstrated to be an effective tool to obtain very good approximations to the solutions of these problems. Our goal, if met, will advance our ability to utilize modern computers to solve these problems of significant importance.
