Mathematical Sciences: Numerical Methods for Hyperbolic Systems

本文将研究具有非齐次性和一般守恒律的双曲型方程组的9404157金数值方法，包括三个主要项目：一、继续研究和发展具有松弛项的双曲型方程组的欠分辨数值格式，并将其应用于稀薄气体动力学的相变模拟和动力学理论；二、继续发展、分析和应用一般守恒律的松弛格式；三、研究可压缩欧拉方程的某些动力学格式的流体动力学极限。如果这些项目成功实施，将使人们对一般守恒定律的各种数值方法有更深入的了解，并为解决与守恒定律有关的问题提供新的数值工具。守恒定律体系是数学家、物理学家和工程师一直在研究的最重要和最困难的物理问题之一。例如，这些问题源于流体动力学、稀薄气体动力学和相变。重要的物理量，如质量、动量和能量，都由非线性偏微分方程组控制。对于这些问题中的大多数，都不可能找到精确的解，并且解的定性行为很难通过数学分析来获得。因此，数值模拟在这些问题的研究中起着至关重要的作用。对这些问题的成功数值模拟不仅需要现代先进的计算设备，而且需要正确、准确和高效的数值方法。这项研究的目标是发展稳健的数值方法，分析它们的行为，并将它们应用于探索由守恒定律系统描述的有趣但具有挑战性的物理问题。

9404157 Jin Numerical methods for hyperbolic systems with inhomogeneities and general conservation laws will be studied, involving three main projects: I. continuing the study and development of under-resolved numerical schemes for hyperbolic systems with relaxations, with applications in phase transition modeling and kinetic theory of rarefied gas dynamics; II. continuing the development, analysis and application of the relaxation schemes for general conservation laws; III. studying the fluid dynamic limits of some kinetic schemes for the compressible Euler equations. These projects, if successfully carried out, will put various numerical approaches for general conservation laws into perspective, and provide new numerical tools for solving problems related to conservation laws. Systems of conservation laws are among the most important and difficult physical problems that mathematicians, physicists and engineers have been investigating. These problems arise from, for example, fluid dynamics, rarefied gas dynamics, and phase transitions. The important physical quantities, such as mass, momentum and energy, are governed by systems of nonlinear partial differential equations. For most of these problems it is impossible to find the exact solutions, and the qualitative behavior of the solutions is exceedingly hard to obtain by mathematical analysis. Thus, numerical simulation has been playing a critical role in studying these problems. Successful numerical simulations to these problems require not only modern advanced computing equipment, but also numerical methods that work correctly, accurately and efficiently. The goal of the proposed research is to develop robust numerical methods, to analyze their behavior, and moreover, to apply them to explore the interesting yet challenging physical problems that are described by systems of conservation laws.