  We perform a linear and entropy stability analysis for wall boundary
condition procedures for discontinuous Galerkin spectral element approximations
of the compressible Euler equations. Two types of boundary procedures are
examined. The first defines a special wall boundary flux that incorporates the
boundary condition. The other is the commonly used reflection condition where
an external state is specified that has an equal and opposite normal velocity.
The internal and external states are then combined through an approximate
Riemann solver to weakly impose the boundary condition. We show that with the
exact upwind and Lax-Friedrichs solvers the approximations are energy
dissipative, with the amount of dissipation proportional to the square of the
normal Mach number. Standard approximate Riemann solvers, namely
Lax-Friedrichs, HLL, HLLC are entropy stable. The Roe flux is entropy stable
under certain conditions. An entropy conserving flux with an entropy stable
dissipation term (EC-ES) is also presented. The analysis gives insight into why
these boundary conditions are robust in that they introduce large amounts of
energy or entropy dissipation when the boundary condition is not accurately
satisfied, e.g. due to an impulsive start or under resolution.
