  We consider the parallel time integration of the linear advection equation
with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms.
Our aim is to develop a better understanding of the convergence behaviour of
these algorithms for this problem, which is known to be poor relative to the
diffusion equation, its model parabolic counterpart. Using Fourier analysis, we
derive new convergence estimates for these algorithms which, in conjunction
with existing convergence theory, provide insight into the origins of this poor
performance. We then use this theory to explore improved coarse-grid
time-stepping operators. For several high-order discretizations of the
advection equation, we demonstrate that there exist non-standard coarse-grid
time stepping operators that yield significant improvements over the standard
choice of rediscretization.
