On the long-time simulation accuracy of the discontinuous Galerkin method for 1D transport equation

Unsteady corrector method is applied to study the DG method for $1\mathrm{D}$ transport equation with constant velocity. Only smooth solutions are considered. On arbitrary non-uniform mesh we prove that the numerical error satisfies $||e|| = O(h^{k+1} + th^{2k+1})$ where $k + 1$ is the number of DOFs at one cell.