Filtering Strategies for Radiation Transport Equations

Radiation is the fundamental mechanism of energy exchange in many physical processes and in the operations of devices for energy and medical applications. In addition, radiation is used as an experimental tool to explore the basic structure of materials. Improved mathematical algorithms and analysis of radiation transport equations will enable important advances in these areas. More generally, these equations serve as a mathematical prototype for a variety of kinetic models that are used to describe dilute gases, plasmas, and multiphase flows. They also demonstrate the fundamental multi-scale nature of driven-dissipative systems. Historically, radiation transport equations have been a driver for many fundamental developments in basic numerical methods research. That trend will continue with this project. On the education side, this project will support and train a student for a career in computational mathematics. This includes support for travel to attend conferences and workshops in order to share research and provide networking opportunities. This grant will support 1 graduate student per year for 3 years.

This project seeks to address one of the fundamental challenges in the kinetic description of radiation: the onset of ray effects in the discrete ordinates approximation of the radiation transport equation. The approach proposed here is based on the use of filters, which smooth the mathematical solution of these equations in a specified manner, in order to improve the fidelity of numerical solutions computed with under-resolved meshes. The project will advance numerical methods for radiation transport equations and, more generally, for kinetic equations and complex multi-physics systems. It will enable to the development of spectral approximation techniques and the synthesis of such techniques into the solution of large scale equations. It will introduce analysis tools to better understand and improve the accuracy of numerical approximation that are under-resolved, and it will establish connections with other types of filtering and sub-scale modeling approaches that are currently used in the simulation of fluids.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.