Numerical Methods for Hyperbolic Systems and Related Problems

--------------------摘要------------------------ DMS-9704957 PI: Shi Jin题目：双曲型系统及相关问题的数值方法我提出研究和设计具有守恒律的双曲型系统及相关问题的数值方法。潜在的物理问题来自流体动力学、稀薄气体动力学和波传播。我计划在三个主要项目上工作：1。包含动力学方程的松弛双曲型系统鲁棒激波捕获方法的发展2. 具有守恒律和Hamilton-Jacobi方程的粘性系统的松弛近似和松弛格式的构造与研究。3. 随机介质中波传播的渐近与数值研究。这些项目如果成功实施，将提供有吸引力的数值方法，能够模拟各种具有挑战性的物理和工业问题。它们还有助于对潜在的物理现象有更多的了解。随着现代计算机的发展，科学计算在科学研究中起着核心作用。我们在这里研究的物理和工业问题出现在复杂的流体、光学和材料中，这些问题是其他科学工具无法解决的。然而，数值计算已被证明是一种有效的工具，可以很好地逼近这些问题的解。我们的目标，如果实现，将提高我们利用现代计算机解决这些重要问题的能力。

-------------------- 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.