  Deterministic solutions of the Boltzmann equation represent a real challenge
due to the enormous computational effort which is required to produce such
simulations and often stochastic methods such as Direct Simulation Monte Carlo
(DSMC) are used instead due to their lower computational cost. In this work, we
show that combining different technologies for the discretization of the
velocity space and of the physical space coupled with suitable time integration
techniques, it is possible to compute very precise deterministic approximate
solutions of the Boltzmann model in different regimes, from extremely rarefied
to dense fluids, with CFL conditions only driven by the hyperbolic transport
term. To that aim, we develop modal Discontinuous Galerkin (DG)
Implicit-Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit-Explicit Linear
Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for
solving the Boltzmann model on multidimensional unstructured meshes. The
solution of the Boltzmann collision operator is obtained through fast spectral
methods, while the transport term in the governing equations is discretized
relying on an explicit shock-capturing DG method on polygonal tessellations in
the physical space. A novel class of WENO-type limiters, based on a shifting of
the moments of inertia for each zone of the mesh, is used to control spurious
oscillations of the DG solution across discontinuities. The order of
convergence is numerically measured for different regimes and found to agree
with the theoretical findings. The new methods are validated considering
two-dimensional benchmark test cases typically used in the fluid dynamics
community. A prototype engineering problem consisting of a supersonic flow
around a NACA 0012 airfoil with space-time-dependent boundary conditions is
also presented for which the pressure coefficients are measured.
