Discontinuous Galerkin solution of the Boltzmann equation in multiple spatial dimensions

Abstract

This thesis focuses on the numerical solution of a kinetic description of small scale dilute gas flows when the Navier-Stokes description breaks down. In particular, it investigates alternative solution techniques for the Boltzmann equation typically used when the Knudsen number (ratio of molecular mean free path to characteristic length scale of flow) exceeds (approximately) 0.1. Alternative solution methods are required because the prevalent Boltzmann solution technique, Direct Simulation Monte Carlo (DSMC), experiences a sharp rise in computational cost as the deviation from equilibrium decreases, such as in low signal flows. To address this limitation, L. L. Baker and N. G. Hadjiconstantinou recently developed a variance reduction technique [5] in which one only simulates the deviation from equilibrium. This thesis presents the implementation of this variance reduction approach to a Runge-Kutta Discontinuous Galerkin finite element formulation in multiple spatial dimensions. Emphasis is given to alternative algorithms for evaluating the advection operator terms, boundary fluxes and hydrodynamic quantities accurately and efficiently without the use of quadrature schemes. The collision integral is treated as a source term and evaluated using the variance-reduced Monte Carlo technique presented in [10, 9]. For piecewise linear (p = 1) and quadratic (p = 2) solutions to the Boltzmann equation in 5 spatial dimensions, the developed algorithms are able to compute the advection operator terms by a factor of 2.35 and 2.73 times faster than an algorithm based on quadrature, respectively; with the computation of hydrodynamic quantities, the overall performance improvement is a factor of 8.5 and 10, respectively.

(cont.) Although the collision integral takes up to 90% or more of the total computation cost, these improvements still provide tangible efficiency advantages in steady-flow calculations in which less expensive transient collision-operator calculation routines are used during a substantial part of the flow development. High order convergence in physical space has been verified by applying the implemented RKDG method on a test problem with a continuous solution. Furthermore, when applied to pressure driven Poiseuille flow through a rectangular channel, the steady state mass flux in the collisionless limit (where exact results exist) agrees within 0.5%, 0.8% and 1.2% of that obtained by Sone and Hasegawa [14] for aspect ratios of 1, 2 and 4 respectively under a spatial resolution of 52 x103 . For Kn = 0.2, 1 and 10, our results agree with those obtained by Sone and Hasegawa [14] from solutions of the linearized Boltzmann-Krook-Welander(BKW) equation by comparing them at an "equivalent" Knudsen number of 1.27Kn [21]. These results validate the implementation and demonstrate the feasibility of the variance-reduced RKDG method for solving the full Boltzmann equation in multiple spatial dimensions. To pursue higher accuracy for this pressure driven flow problem, a p = 1 scheme was found to be more efficient than a p = 2 scheme at a coarser spatial discretization. This can be achieved by using finer spatial discretization and non-uniform spacing to generate more elements near regions of discontinuities or large variations in the molecular distribution function.