RUI: Multidimensional Conservation Laws

California State University-Long Beach Foundation, Long Beach CA

Investigators

Abstract

Conservation laws are fundamental laws of nature that govern many phenomena observed in physics and fluid mechanics, as well as in engineering applications. The first part of this research project addresses the mathematical modeling and analysis of systems of multidimensional conservation laws that mostly relate to problems in dynamics of gases and liquids. It focuses on change of type (transonic) problems, from supersonic to subsonic, or mixed type problems with discontinuities, such as vortex waves and shock waves. The commonly known manifestation of the latter is generation of a sonic boom when an airplane exceeds the velocity of sound. This project aims at developing systematic theories to understand the solution structures of these transonic problems in multidimensional conservation laws. The second part of the project aims to investigate the feasibility of various wildfire spread models with sparse data, and develop efficient algorithms to perform simulations for the model problems. In recent years wildfires have become an all too frequent occurrence, especially in the Western United States. The research on the wildfire spread models will enable effective fire-fighting planning, and thus have a direct impact on the welfare of society. The project will take place at a large, urban, Hispanic-serving institution and involve undergraduate/master students in simulations of the proposed problems, preparing them for further work in the design, implementation, and development of the algorithms. This project addresses long standing open problems in multidimensional conservation laws, such as Mach shock reflections to resolve the von Neumann paradox, slip line discontinuity propagation to understand vortex waves, and the transonic flow to study a flow passing an airfoil. The investigator will focus on these nonlinear transonic problems to gain new physical insights, to develop novel analytical tools, and to find the correct mathematical framework in which to pose the nonlinear conservation laws and to perhaps develop efficient numerical methods. This research aims to provide more efficient and effective methods for applications, including compressible gas dynamics, thermodynamics, multi-phase flow, and porous medium flow. A part of this research will be devoted to modeling wildfire spread with reaction-advection-diffusion systems. The investigator will investigate the feasibility of various wildfire spread models with sparse data and develop efficient algorithms to solve the model problems. Results will be tested on realistic data. Some aspects of this project will be conducted in collaboration with early career researchers, and in communication with the USDA Forest Fire Lab in Riverside, CA.