  Parallel-in-time algorithms have been successfully employed for reducing
time-to-solution of a variety of partial differential equations, especially for
diffusive (parabolic-type) equations. A major failing of parallel-in-time
approaches to date, however, is that most methods show instabilities or poor
convergence for hyperbolic problems. This paper focuses on the analysis of the
convergence behavior of multigrid methods for the parallel-in-time solution of
hyperbolic problems. Three analysis tools are considered that differ, in
particular, in the treatment of the time dimension: (1) space-time local
Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic
mode analysis, coupling standard local Fourier analysis approaches in space
with algebraic computation in time, and (3) a two-level reduction analysis,
considering error propagation only on the coarse time grid. In this paper, we
show how insights from reduction analysis can be used to improve feasibility of
the semi-algebraic mode analysis, resulting in a tool that offers the best
features of both analysis techniques. Following validating numerical results,
we investigate what insights the combined analysis framework can offer for two
model hyperbolic problems, the linear advection equation in one space dimension
and linear elasticity in two space dimensions.
