Mathematical Sciences: Shock Waves and Conservation Laws

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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.