Dynamical low-rank methods for capturing kinetic effects in the collisional transition regime

Kinetic equations such as the Boltzmann-Maxwell system describe plasma behavior

in collisionless or only moderately collisional regimes, where fluid models

may not be appropriate. For example, recent work (Vogman et. al, Phys. Plasmas 2020) has

demonstrated the presence of kinetic effects, including finite Larmor radius

effects, in the weakly collisional, magnetized Kelvin Helmholtz instability.

Unfortunately, the curse of dimensionality makes the numerical solution

of 5 or 6-dimensional kinetic equations extremely costly. Recently,

developments in the method of dynamical low-rank approximation (DLRA) have begun to

chart a path around the curse of dimensionality for kinetic equations. We

summarize the DLRA method and its application to

the collisional Boltzmann-Maxwell equation. Several modifications to the basic

formulation, made with the aim of preserving structure such as conservation and Maxwellian

equilibrium, are presented. We pay special attention to achieving

computational speedups in the collisional transition regime, characterized

by Knudsen numbers in the range 10^-2 < Kn < 10.