Jeffrey R. Haack — Research Statement — Fall 2012

My research focuses on scientific computing and kinetic equations. More specifically I develop and analyze numerical methods for solving the Boltzmann transport equation and similar kinetic models, as well as their application to high performance computing resources. My work is supported by NSF-DMS Grant 1217254 “Accurate high performance computing for nonlinear collisional kinetic theory”, (2012-2015, award amount $167,677). Kinetic equations describe physics at the mesoscopic level, where there are large numbers of interactions between particles but they are not in thermodynamic equilbrium and thus cannot be described by continuum dynamics. These types of equations classically arose from statistical physics for modeling rarefied gases and plasmas, most famously through the Boltzmann transport equation. More recently, kinetic models have been applied to describe biological and social processes such as aggregation and collective movement in animals, crowd dynamics of pedestrians, and financial markets. The Boltzmann equation [13, 14] is an integrodifferential equation that describes the evolution of a dilute gas of particles where only binary interactions are considered. For elastic collisions, the microscopic interaction, or collision, between particles is required to perserve the mass, momentum, and energy before and after the interaction. Under these constraints, two colliding particles with velocity v and v∗ will result in the post-collisional velocities v ′, v∗ given by: v′ = v + v∗ 2 + |u|σ 2 , v∗ = v + v∗ 2 − |u|σ 2 ,