  Time-parallel algorithms seek greater concurrency by decomposing the temporal
domain of a Partial Differential Equation (PDE), providing possibilities for
accelerating the computation of its solution. While parallelisation in time has
allowed remarkable speed-ups in applications involving parabolic equations, its
effectiveness in the hyperbolic framework remains debatable: growth of
instabilities and slow convergence are both strong issues in this case, which
often negate most of the advantages from time-parallelisation. Here, we focus
on the Multigrid Reduction in Time (MGRIT) algorithm, investigating in detail
its performance when applied to non-linear conservation laws with a variety of
discretisation schemes. Specific attention is given to high-accuracy Weighted
Essentially Non-Oscillatory (WENO) reconstructions, coupled with Strong
Stability Preserving (SSP) integrators, which are often the discretisations of
choice for such PDEs. A technique to improve the performance of MGRIT when
applied to a low-order, more dissipative scheme is also outlined. This study
aims at identifying the main causes for degradation in the convergence speed of
the algorithm, and finds the Courant-Friedrichs-Lewy (CFL) limit to be the
principal determining factor.
