  In this paper, we study superconvergence properties of the local
discontinuous Galerkin method for one-dimensional linear parabolic equations
when alternating fluxes are used. We prove, for any polynomial degree $k$, that
the numerical fluxes converge at a rate of $2k+1$ (or $2k+1/2$) for all mesh
nodes and the domain average under some suitable initial discretization.
  We further prove a $k+1$th superconvergence rate for the derivative
approximation and a $k+2$th superconvergence rate for the function value
approximation at the Radau points.
  Numerical experiments demonstrate that in most cases, our error estimates are
optimal, i.e., the error bounds are sharp.
