Mathematical Sciences: Shock Waves and Conservation Laws

本研究解决了在激波存在时求解非线性双曲守恒律的理论和计算方法研究的重要问题。研究的主要目的是了解非线性波及其在物理过程中的相互作用，以及模拟这些过程的数值方法。主题包括研究波形的稳定性和由于波速重合而表现出非线性共振的系统中数值方法的稳定性；构建非均匀系统或守恒定律中共振波相互作用的一般理论（包括变直径管道中跨音速气体动力学流动的系统）；并从非线性波的角度研究广义相对论中的重力波。这个项目属于应用数学领域，涉及冲击波的研究。其研究成果涉及非线性偏微分方程、泛函分析、几何、数值分析、科学计算和数学物理等领域。

This research addresses issues important to the study of theoretical and computational methods of obtaining solutions to nonlinear hyperbolic conservation laws when shock waves are present. The main objective of the research is to understand nonlinear waves and their interactions in physical processes, and in numerical methods that simulate these processes. Topics include a study of the stability of wave patterns and the stability of numerical methods in systems exhibiting nonlinear resonance due to coincidence of wave speeds; a program for constructing a general theory of resonant wave interactions in inhomogeneous systems or conservation laws (including systems that arise in transonic gas dynamical flow in variable diameter pipelines); and the study of gravity waves in general relativity from the point of view of nonlinear waves. The project is in the area of applied mathematics dealing with the study of shock waves. The results of this research impinge upon the fields of nonlinear partial differential equations, functional analysis, geometry, numerical analysis, scientific computing, and mathematical physics.