  This paper investigates superconvergence properties of the local
discontinuous Galerkin methods with generalized alternating fluxes for
one-dimensional linear convection-diffusion equations. By the technique of
constructing some special correction functions, we prove the $(2k+1)$th order
superconvergence for the cell averages, and the numerical traces in the
discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are
obtained for the error and its derivative at generalized Radau points. All
theoretical findings are confirmed by numerical experiments.
