This project consists of two parts. The first project aims at understanding the solution structures of mixed (hyperbolic-elliptic) type systems of partial differential equations arising from multidimensional conservation laws. A distinctive feature of multidimensional conservation laws written in self-similar coordinates is that they change their type: they are hyperbolic far from the origin, and mixed near the origin. The latter confronts us with important problems in nonlinear equations of mixed types and free boundaries. In particular, when the waves are weak enough that the nonlinear acoustic waves dominate the nonlinear entropy and vorticity waves, shock polar analysis fails to explain the nature of shock reflection. This is the so called von Neumann paradox. It is one of the main motivations of this project to address the failures of asymptotic and computational analysis to resolve certain paradoxes concerning the existence and the stability of solutions of multidimensional problems. Understanding the mathematical structure of multidimensional conservation laws is a crucial step in improving computational methods and in resolving such paradoxes. The project is directed at investigating these mixed type 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 develop efficient numerical methods. The second project will investigate the feasibility of various wildfire spread models with sparse data, and development of efficient algorithms to solve the model problems. This project will be conducted in communication with the USDA Forest Fire Lab in Riverside, CA. The wildfire spread models inherit scale separation of the local atmospheric dynamics (one kilometer and larger) and the local combustion dynamics (one meter and smaller). Also, available data are irregular and sometimes inaccurate. Existing extensive models, require complete data, are computationally intensive, and thus may not provide immediate results, which are crucial for effective fire-fighting plans. The project will develop efficient algorithms for simplified models, which can be incorporated with sparse data, so that the fire-fighting could be planned immediately and as accurately as possible.
Multidimensional conservation laws are mathematical models for fundamental processes in physics and engineering, such as high-speed flows and supersonic jets. A deeper understanding of multidimensional conservation laws will provide efficient and effective methods for applications, such as compressible gas dynamics, thermodynamics, multi-phase flow and porous medium flow. Our wildfire modeling will contribute to effective fire-fighting planning and thus will have direct benefits for society. This project will take place at a Hispanic-serving institution, and involve undergraduate/master students in simulations of the model problems, thus preparing the students for further work in the design, implementation, and development of algorithms. This project will provide students with training to prepare them for their academic careers (as Ph.D. students), or their future jobs in the high tech industry in the greater Los Angeles area.