Title: Wave propagation characteristics of Parareal,
Abstract: The paper derives and analyses the (semi-)discrete dispersion relation of the
Parareal parallel-in-time integration method. It investigates Parareal's wave
propagation characteristics with the aim to better understand what causes the
well documented stability problems for hyperbolic equations. The analysis shows
that the instability is caused by convergence of the amplification factor to
the exact value from above for medium to high wave numbers. Phase errors in the
coarse propagator are identified as the culprit, which suggests that
specifically tailored coarse level methods could provide a remedy.