Efficient numerical methods for solving the Boltzmann equation for small scale flows

Abstract

The Navier-Stokes equations of continuum fluid mechanics fail to accurately describe dilute gas flows when the characteristic lengthscale of the system is on the order of (or smaller than) the molecular mean free path. At these lengthscales, gaseous hydrodynamics may be described by a kinetic description, namely the Boltzmann equation. Currently, the prevalent method for solving the Boltzmann equation is a particle simulation method known as direct simulation Monte Carlo (DSMC). DSMC is very efficient for high-speed (more generally, high signal) flows; unfortunately, due to the statistical sampling used to obtain hydrodynamic fields, the computational cost of DSMC (for a given signal to noise ratio) increases rapidly with decreasing signal. For example, the computational cost for calculating the flow velocity with a fixed signal to noise ratio scales with Ma-2 as Ma -- 0 (Ma is the Mach number). As a result, simulation of many low-signal flows of practical interest (for example, in micro- and nano-scale devices) is currently not feasible using DSMC. This thesis describes how the above limitation can be alleviated through the use of variance reduction techniques. In particular, we show that by simulating only the deviation from equilibrium, one can devise a variety of numerical methods that have a computational cost that is both small and independent of the magnitude of this deviation.

(cont.) For low-speed flows, this leads to methods that are significantly more efficient than DSMC. Two implementations of this variance reduction concept are presented. The first is a particle method akin to DSMC, differing only in ways necessary to simulate the deviation from equilibrium. This particle formulation retains the most important strengths of DSMC - specifically, importance sampling (providing computational efficiency) and the ability to capture discontinuities in the solution - while offering a significant computational advantage compared to DSMC for low-signal flows. The second approach considered is a PDE-based method using a discontinuous Galerkin formulation, which is able to treat traveling discontinuities. This PDE-based approach has the potential for high-order accuracy, as well as implicit steady-state formulations which can be significantly more efficient when transient phenomena are not of interest.