Mathematical Sciences: Numerical Methods for Hyperbolic Systems

小行星9404157 本文将研究具有非齐次项和一般守恒律的双曲型方程组的数值方法，主要包括三个方面的内容：I。继续研究和开发松弛双曲系统的欠分辨数值方案，并应用于稀薄气体动力学的相变建模和动力学理论; II.继续发展、分析和应用一般守恒定律的松弛方案;三.研究了可压缩欧拉方程某些动力学格式的流体动力学极限。 这些项目，如果成功地进行，将把各种数值方法的一般守恒律的角度来看，并提供新的数值工具，解决问题的守恒律。 守恒律系统是数学家、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。 这些问题产生于例如流体动力学、稀薄气体动力学和相变。 重要的物理量，如质量，动量和能量，是由非线性偏微分方程组。 对于大多数这些问题是不可能找到精确的解决方案，解决方案的定性行为是非常难以获得的数学分析。 因此，数值模拟在研究这些问题中起着至关重要的作用。 成功地对这些问题进行数值模拟，不仅需要现代先进的计算设备，而且需要正确、准确、高效的数值方法。 拟议研究的目标是开发强大的数值方法，分析它们的行为，并应用它们来探索守恒律系统所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.