This project aims to build an integrated program of research and education focused on advances in predictive simulations of complex kinetic systems. Such systems are comprised of a large number of particles in random motion and are best described by the Boltzmann and related kinetic equations. In practical applications, there are many sources of uncertainties that can arise in kinetic systems: imprecise measurements for initial and boundary conditions, incomplete knowledge of the fundamental interaction mechanism between particles, and so on. Understanding the impact of these uncertainties is critical to the simulations of the complex kinetic systems, and will allow scientists and engineers to obtain more reliable predictions and perform better risk assessment. Due to the unique challenges arising in kinetic equations, such as multiple scales, high dimensionality, and positivity, very few existing generic uncertainty quantification (UQ) algorithms can be applied directly. To bridge this gap, the research objective of this project is to develop highly efficient stochastic and multiscale numerical methods for Boltzmann-like kinetic equations. A parallel educational objective is to create innovative opportunities for students at all levels to improve science, technology, engineering, and mathematics (STEM) education and promote career interest in these disciplines, especially among female students.
Specifically, we will pursue four research and educational aims: 1) develop stochastic asymptotic-preserving methods for multiscale kinetic equations; 2) construct high performance stochastic algorithms for the Boltzmann collision operator; 3) design physics-preserving UQ algorithms for kinetic systems; and 4) create education and outreach activities for students through undergraduate STEM classroom innovation; graduate curriculum development in kinetic theory; graduate and undergraduate mentoring; and organizing an after-school math research program for high school girls and family math/science nights at middle and elementary schools.
