  This work presents and compares efficient implementations of high-order
discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG)
method, a hybridizable discontinuous Galerkin (HDG) method, and a primal
formulation of HDG, applied to the implicit solution of unsteady compressible
flows. The matrix-free implementation allows for a reduction of the memory
footprint of the solver when dealing with implicit time-accurate
discretizations. HDG reduces the number of globally-coupled degrees of freedom
relative to DG, at high order, by statically condensing element-interior
degrees of freedom from the system in favor of face unknowns. The primal
formulation further reduces the element-interior degrees of freedom by
eliminating the gradient as a separate unknown. This paper introduces a
$p$-multigrid preconditioner implementation for these discretizations and
presents results for various flow problems. Benefits of the $p$-multigrid
strategy relative to simpler, less expensive, preconditioners are observed for
stiff systems, such as those arising from low-Mach number flows at high-order
approximation. The $p$-multigrid preconditioner also shows excellent
scalability for parallel computations. Additional savings in both speed and
memory occur with a matrix-free/reduced version of the preconditioner.
