  We compare high-order methods including spectral difference (SD), flux
reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element
method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the
best candidate to simulate strong shock waves characteristic of hypersonic
flows. We consider several benchmarks, including the Leblanc and modified
shock-density wave interaction problems that require robust stabilization and
positivity-preserving properties for a successful flow realization. We also
perform simulations of the three-species Sod problem with simplified chemistry
with the chemical reaction source terms introduced in the Euler equations. The
ES-DGSEM scheme exhibits the highest stability, negligible numerical
oscillations, and requires the least computational effort in resolving reactive
flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to
hypersonic Euler equations by deriving a new set of two-point entropy
conservative fluxes for a five-species gas model. Stabilization for capturing
strong shock waves occurs by blending high-order entropy conservative fluxes
with low-order finite volume fluxes constructed using the HLLC Riemann solver.
The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod
problem. To this end, we adopt the Mutation++ library to compute the reaction
source terms, thermodynamic properties, and transport coefficients. We also
investigate the effect of real chemistry versus ideal chemistry, and the
results demonstrate that the ideal chemistry assumption fails at high
temperatures, hence real chemistry must be employed for accurate predictions.
Finally, we consider a viscous hypersonic flow problem to verify the transport
coefficients and reaction source terms determined by the Mutation++ library.
