### A Parallel, Matrix-free Newton Method for Solving Approximate Boltzmann Equations on Unstructured Topologies

*A Dissertation Presented for the Doctor of Philosophy in Computational Engineering, The
University of Tennessee at Chattanooga*

#### Robert Glenn Brook, December 2008

Abstract:

This work presents a parallel algorithm for the numerical solution of the Boltzmann
equation subject to approximation of the collision integral by either the Bhatnagar-Gross-Krook
(BGK) collision model or the Shakhov collision model. The algorithm employs the discrete
ordinate method in conjunction with the composite trapezoidal quadrature rule to remove
the velocity space dependence of the distribution function, casting the approximate
Boltzmann equation into a set of linear advection equations with nonlinear source
terms defined across physical space. A matrix-free Newton-Symmetric-Gauss-Seidel method
based on the machine-accurate calculation of Jacobian-vector products using complex
Taylor series approximations is employed to solve the resulting set of equations using
a fully implicit, node-centered, finite volume formulation on multi-element, unstructured
grids. The parallel implementation is based on a domain decomposition across physical
space in which each processing unit is assigned a single subdomain. Communications
between processing units are implemented using library routines specified by the Message
Passing Interface (MPI) standard. Proper parallel performance, as evidenced by speedup,
is verified; and, grid refinement studies in both physical space and velocity space
are conducted to verify the algorithmic order of accuracy. Validation results are
generated for plane Couette flows, normal shock waves, and an unsteady shock tube.
Finally, the algorithm is applied to model gas flows in a two-dimensional microchannel
and a Knudsen compressor channel in two and three dimensions.