Discontinuous Galerkin solution of the Boltzmann equation in multiple spatial dimensions

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Computation for Design and Optimization Program.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2007

Author dc:creator

• Lian, Zhengyi

Advisor dc:contributor.advisor

• Nicholas G. Hadjiconstantinou.

• Computation for Design and Optimization Program.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/41560

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/41560