  In this paper, we study superconvergence properties of the discontinuous
Galerkin (DG) method for one-dimensional linear hyperbolic equation when upwind
fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or
$2k+1/2$th) superconvergence rate of the DG approximation at the downwind
points and for the domain average under quasi-uniform meshes and some suitable
initial discretization. Moreover, we prove that the derivative approximation of
the DG solution is superconvergent with a rate $k+1$ at all interior left Radau
points. All theoretical finding are confirmed by numerical experiments.
