  The Boltzmann equation, an integro-differential equation for the molecular
distribution function in the physical and velocity phase space, governs the
fluid flow behavior at a wide range of physical conditions, including
compressible, turbulent, as well as flows involving further physics such as
non-equilibrium internal energy exchange and chemical reactions. Despite its
wide applicability, deterministic solution of the Boltzmann equation presents a
huge computational challenge, and often the collision operator is simplified
for practical reasons. In this work, we introduce a highly accurate
deterministic method for the full Boltzmann equation which couples the
Runge-Kutta discontinuous Galerkin (RKDG) discretization in time and physical
space (Su et al., Comp. Fluids, 109 pp. 123-136, 2015) and the recently
developed fast Fourier spectral method in velocity space (Gamba et al., SIAM J.
Sci. Comput., 39 pp.~B658--B674, 2017). The novelty of this approach
encompasses three aspects: first, the fast spectral method for the collision
operator applies to general collision kernels with little or no practical
limitations, and in order to adapt to the spatial discretization, we propose
here a singular-value-decomposition based algorithm to further reduce the cost
in evaluating the collision term; second, the DG formulation employed has high
order of accuracy at element-level, and has shown to be more efficient than the
finite volume method; thirdly, the element-local compact nature of DG as well
as our collision algorithm is amenable to effective parallelization on
massively parallel architectures. The solver has been verified against
analytical Bobylev-Krook-Wu solution. Further, the standard benchmark test
cases of rarefied Fourier heat transfer, Couette flow, oscillatory Couette
flow, normal shock wave, lid-driven cavity flow, and thermally driven cavity
flow have been studied.
