  We investigate two efficient time discretizations for the post-processing
technique of discontinuous Galerkin (DG) methods to solve hyperbolic
conservation laws. The post-processing technique, which is applied at the final
time of the DG method, can enhance the accuracy of the original DG solution
(spatial superconvergence). One main difficulty of the post-processing
technique is that the spatial superconvergence after post-processing needs to
be matched with proper temporary accuracy. If the semi-discretized system (ODE
system after spatial discretization) is under-resolved in time, then the space
superconvergence will be concealed. In this paper, we focus our investigation
on the recently designed SDG method and derive its explicit scheme from a
correction process based on the DG weak formulation. We also introduce another
similar technique, namely the spectral deferred correction (SDC) method. A
comparison is made among both proposed time discretization techniques with the
standard third-order Runge-Kutta method through several numerical examples, to
conclude that both the SDG and SDC methods are efficient time discretization
techniques for exploiting the spatial superconvergence of the DG methods.
