This research project develops mathematical tools and numerical algorithms to study multiscale problems governed by a selected set of partial differential equations arising in diverse applications. The proposed work consists of three projects: recovery of high frequency wave fields governed by a variety of wave equations, including hyperbolic equations such as the acoustic wave equation, dispersive equations such as the Schroedinger equation, and the time independent Helmholtz equation; development of kinetic theory of photons in the transport of energy; and design of entropy satisfying discontinuous Galerkin methods for kinetic Fokker-Planck equations, with emphasis on kinetic models arising in polymeric fluids and collective motions in biology.
Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Recovery of high frequency wave fields is a fundamental problem in high frequency wave propagation, the study of which can provide a deeper understanding of high frequency wave dynamics occurring in various applications. The study of condensation of photons would result in methods which may potentially be suitable for designing novel light sources. Entropy satisfying methods are important in capturing the right physics in long time numerical simulations. The design of such methods has the potential to elucidate many aspects of the physical process. The theory will be applied to and driven by identified practical applications.
