  We present new results building on the conservative deterministic spectral
method for the space inhomogeneous Boltzmann equation developed by Gamba and
Tharkabhushaman. This approach is a two-step process that acts on the weak form
of the Boltzmann equation, and uses the machinery of the Fourier transform to
reformulate the collisional integral into a weighted convolution in Fourier
space. A constrained optimization problem is solved to preserve the mass,
momentum, and energy of the resulting distribution. We extend this method to
second order accuracy in space and time, and explore how to leverage the
structure of the collisional formulation for high performance computing
environments. The locality in space of the collisional term provides a
straightforward memory decomposition, and we perform some initial scaling tests
on high performance computing resources. We also use the improved computational
power of this method to investigate a boundary-layer generated shock problem
that cannot be described by classical hydrodynamics.
