  We present an analysis of stationary discrete shock profiles for a
discontinuous Galerkin method approximating scalar nonlinear hyperbolic
conservation laws with a convex flux. Using the Godunov method for the
numerical flux, we characterize the steady state solutions for arbitrary
approximation orders and show that they are oscillatory only in one mesh cell
and are parametrized by the shock strength and its relative position in the
cell. In the particular case of the inviscid Burgers equation, we derive
analytical solutions of the numerical scheme and predict their oscillations up
to fourth-order of accuracy. Moreover, a linear stability analysis shows that
these profiles may become unstable at points where the Godunov flux is not
differentiable. Theoretical and numerical investigations show that these
results can be extended to other numerical fluxes. In particular, shock
profiles are found to vanish exponentially fast from the shock position for
some class of monotone numerical fluxes and the oscillatory and unstable
characters of their solutions present strong similarities with that of the
Godunov method.
