In this paper we study a class of one-sided post-processing techniques
to enhance the accuracy of the discontinuous Galerkin methods. The
applications considered in this paper are linear hyperbolic equations,
however the technique can be used for the solution to a discontinuous
Galerkin method solving other types of partial differential equations,
or more general approximations, as long as there is a higher order
negative norm error estimate for the numerical solution. The advantage
of the one-sided post-processing is that it uses information only from
one side, hence it can be applied up to domain boundaries, a discontinuity
in the solution, or an interface of different mesh sizes. This technique
allows us to obtain an improvement in the order of accuracy from k+1 of
the discontinuous Galerkin method to 2k+1 of the post-processed solution,
using piecewise polynomials of degree k, throughout the entire domain
and not just away from the boundaries, discontinuities, or interfaces of
different mesh sizes.