  We consider the time discretization of a linear parabolic problem by the
discontinuous Galerkin (DG) method using piecewise polynomials of degree at
most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known
that the spatial $L_2$-norm of the DG error is of optimal order $k^r$ globally
in time, and is, for $r\ge2$, superconvergent of order $k^{2r-1}$ at the nodes.
We show that on the $n$th subinterval $(t_{n-1},t_n)$, the dominant term in the
DG error is proportional to the local right Radau polynomial of degree $r$.
This error profile implies that the DG error is of order $k^{r+1}$ at the
right-hand Gauss--Radau quadrature points in each interval. We show that the
norm of the jump in the DG solution at the left end point $t_{n-1}$ provides an
accurate \emph{a posteriori} estimate for the maximum error over the
subinterval $(t_{n-1},t_n)$. Furthermore, a simple post-processing step yields
a \emph{continuous} piecewise polynomial of degree $r$ with the optimal global
convergence rate of order $k^{r+1}$. We illustrate these results with some
numerical experiments.
