  In this paper we present a numerical method for the Boltzmann equation. It is
a spectral discretization in the velocity and a discontinuous Galerkin
discretization in physical space. To obtain uniform approximation properties in
the mach number, we shift the velocity by the (smoothed) bulk velocity and
scale it by the (smoothed) temperature, both extracted from the density
distribution. The velocity trial functions are polynomials multiplied by a
Maxwellian. Consequently, an expansion with a low number of trial functions
already yields satisfying approximation quality for nearly equilibrated
solutions. By the polynomial test space, density, velocity and energy are
conserved on the discrete level. Different from moment methods, we stabilize
the free flow operator in phase space with upwind fluxes. Several numerical
results are presented to justify our approach.
