  In this paper, we use Fourier analysis to study the superconvergence of the
semi-discrete discontinuous Galerkin method for scalar linear advection
equations in one spatial dimension. The error bounds and asymptotic errors are
derived for initial discretization by $L_2$ projection, Gauss-Radau projection,
and other projections proposed by Cao et. al. For pedagogical purpose, the
errors are computed in two different ways. In the first approach, we compute
the difference between the numerical solution and a special interpolation of
the exact solution, and show that it consists of an asymptotic error of order
$2k+1$ and a transient error of lower order. In the second approach, as by
Chalmers and Krivodonova, we compute the error directly by decomposition into
physical and nonphysical modes, and obtain agreement with the first approach.
We then extend the analysis to vector conservation laws, solved using the
Lax-Friedrichs flux. We prove that the superconvergence holds with the same
order. The error bounds and asymptotic errors are demonstrated by various
numerical experiments for scalar and vector advection equations.
