Efficient numerical methods for solving the Boltzmann equation for low-speed flows

Abstract

When the Knudsen number, typically defined as the ratio of the molecular mean free path to the characteristic length scale of a dilute gas flow, is larger than approximately 0.1, the Navier-Stokes equations are no longer valid. In this case, which is frequently encountered in small-scale flows, one must solve the more general Boltzmann equation. The objective of this work is to develop a method which requires a lower computational cost than existing methods for low speed flows. This thesis describes and analyzes the performance of a method to solve the Boltzmann equation for dilute gas flows by a direct numerical method rather than by the more prevalent stochastic molecular simulation approach. In this work, the evaluation of the collision integral of the Boltzmann equation is performed using a quasi-random Monte Carlo integration approach for faster convergence. In addition, interpolation is used to reduce the effect of discretization errors. We find that cubic interpolation leads to accurate solutions which exhibit excellent conservation properties, thus eliminating the need for an artificial correction step. The use of quasi-random sequences is shown to provide a significant speedup, which increases as the discretization becomes finer. For the problems investigated here, the maximum speedup observed is on the order of four.