  Simulations of the discrete Boltzmann Bhatnagar-Gross-Krook (BGK) equation
are an important tool for understanding fluid dynamics in non-continuum
regimes. Here, we introduce a discontinuous Galerkin finite element method
(DG-FEM) for spatial discretization of the discrete Boltzmann equation for
isothermal flows with Knudsen numbers (Kn~O(1)). In conjunction with a
high-order Runge-Kutta time marching scheme, this method is capable of
achieving high-order accuracy in both space and time, while maintaining a
compact stencil. We validate the spatial order of accuracy of the scheme on a
two-dimensional Couette flow with Kn = 1 and the D2Q16 velocity discretization.
We then apply the scheme to lid-driven micro-cavity flow at Kn = 1, 2, and 8,
and we compare the ability of Gauss-Hermite (GH) and Newton-Cotes (NC) velocity
sets to capture the high non-linearity of the flow-field. While GH quadrature
provides higher integration strength with fewer points, the NC quadrature has
more uniformly distributed nodes with weights greater than machine-zero,
helping to avoid the so-called ray-effect. Broadly speaking, we anticipate that
the insights from this work will help facilitate the efficient implementation
and application of high-order numerical methods for complex high Knudsen number
flows.
