  In the spirit of making high-order discontinuous Galerkin (DG) methods more
competitive, researchers have developed the hybridized DG methods, a class of
discontinuous Galerkin methods that generalizes the Hybridizable DG (HDG), the
Embedded DG (EDG) and the Interior Embedded DG (IEDG) methods. These methods
are amenable to hybridization (static condensation) and thus to more
computationally efficient implementations. Like other high-order DG methods,
however, they may suffer from numerical stability issues in under-resolved
fluid flow simulations. In this spirit, we introduce the hybridized DG methods
for the compressible Euler and Navier-Stokes equations in entropy variables.
Under a suitable choice of the stabilization matrix, the scheme can be shown to
be entropy stable and satisfy the Second Law of Thermodynamics in an integral
sense. The performance and robustness of the proposed family of schemes are
illustrated through a series of steady and unsteady flow problems in subsonic,
transonic, and supersonic regimes. The hybridized DG methods in entropy
variables show the optimal accuracy order given by the polynomial approximation
space, and are significantly superior to their counterparts in conservation
variables in terms of stability and robustness, particularly for under-resolved
and shock flows.
