Efficient particle methods for solving the Boltzmann equation

Abstract

A new particle simulation method for solving the Boltzmann equation is presented and tested. This method holds a significant computational efficiency advantage for low-signal flows compared to traditional particle methods such as the Direct Simulation Monte Carlo (DSMC). More specifically, the proposed algorithm can efficiently simulate arbitrarily small deviations from equilibrium (e.g. low speed flows) at a computational cost that does not scale with the deviation from equilibrium, while maintaining the basic algorithmic structure of DSIMC. This is achieved by incorporating the variance reduction ideas presented in [L. L. Baker and N. G. Hadjiconstantinou, Physics of Fluids, vol 17, art. no 051703, 2005] within a collision integral formulation; the latter ensures that the deviation from equilibrium remains finite and thus the calculation remains stable for collision dominated flows, in contrast to previous attempts. The formulation, developed within this thesis, is described in detail. The resulting scheme is validated for a wide range of Knudsen numbers (ratio of molecular mean free path to characteristic flow lengthscale) -- ranging from collision-dominated flow -- to collisionless flow- and a wide range of deviations from equilibrium. Excellent agreement is found with DSMC solutions for linear and weakly non-linear flows.