  This paper presents a numerical approximation technique for the Boltzmann
equation based on a moment system approximation in velocity dependence and a
discontinuous Galerkin finite-element approximation in position dependence. The
closure relation for the moment systems derives from minimization of a suitable
{\phi}-divergence. This divergence-based closure yields a hierarchy of
tractable symmetric hyperbolic moment systems that retain the fundamental
structural properties of the Boltzmann equation. The resulting combined
discontinuous Galerkin moment method corresponds to a Galerkin approximation of
the Boltzmann equation in renormalized form. We present a new class of
numerical flux functions, based on the underlying renormalized Boltzmann
equation, that ensure entropy dissipation of the approximation scheme.
Numerical results are presented for a one-dimensional test case.
