Mathematical Sciences: Numerical Methods for Hyperbolic Systems

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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.