A Progress Report on Numerical Methods for BGK-Type Kinetic Equations

In this report we review some preliminary work on the numerical solution of BGK-type kinetic equations of particle transport. Such equations model the motion of fluid particles via a density field when the kinetic theory of rarefied gases must be used in place of the continuum limit Navier-Stokes and Euler equations. The BGK-type equations describe the fluid in terms of phase space variables, and, in three space dimensions, require 6 independent phase-space variables (3 for space and 3 for velocity) for accurate simulation. This requires sophisticated numerical algorithms and efficient code to realize predictions over desired space and time scales. In particular, stable numerical methods must be designed to handle potential discontinuities (shocks) and rarefaction waves in the solutions coming from conservative advection terms and, in addition, numerical stiffness owing to diffusive particle collision terms. Furthermore, the particle interaction terms are non-local in nature, adding yet another layer of complexity, and the interaction length scales of the non-local terms may be orders of magnitude different, when multiple particle species are involved. In this report, we outline strategies for generating efficient and stable numerical algorithms and code, including the use of (i) stable high-order finite volume methods, (ii) fully implicit and implicit-explicit (IMEX) time integration techniques, and (iii) adaptive time-phase-space multi-level methods. The preliminary codes, which will be demonstrated herein, are built in the commercial software package MATLAB for quick and easy prototyping, but will later be translated into production software using modern open languages

Asymptotic Relaxation of Moment Equations for a Multi-Species, Homogeneous BGK Model

Multi-species BGK models describe the dynamics of rarefied gases with constituent particles of different elements or compounds with potentially non-trivial velocity distributions. In this paper, moment equations for the bulk velocities, energies, and temperatures of a spatially homogeneous multi-species BGK model are examined. A key challenge in analyzing these equations is the fact that the collision frequencies are allowed to depend on the species temperatures, which allows for more realistic simulations of dilute gas flow. Therefore, a positive lower bound is established for the species temperatures. With this lower bound, a global existence and uniqueness of solutions to the coupled velocity-energy ODE system is established. The lower bound also enables a proof of exponential decay to a unique steady-state solution. Numerical results are presented to demonstrate how the bulk velocities and temperatures relax for large times.Comment: 21 pages, 3 figure