Angel, JudithJudithAngelGötschel, SebastianSebastianGötschelRuprecht, DanielDanielRuprecht2026-08-282026-08-282024-05Go20 Conference on Scientific Computing and Software, Go20 CSCS 2024 (2026)https://hdl.handle.net/11420/64540The Parareal parallel-in-time integration method often performs poorly when applied to hyperbolic partial differential equations. This effect is even more pronounced when the coarse propagator uses a reduced spatial resolution. However, some combinations of spatial discretization and numerical time stepping nevertheless allow for Parareal to converge with monotonically decreasing errors. This raises the question how these configurations can be distinguished theoretically from those where the error initially increases, sometimes over many orders of magnitude. For linear problems, we prove a theorem that implies that the 2-norm of the Parareal iteration matrix is not a suitable tool to predict convergence for hyperbolic problems when spatial coarsening is used. We then show numerical results that suggest that the pseudo-spectral radius can reliably indicate if a given configuration of Parareal will show transient growth or monotonic convergence. For the studied examples, it also provides a good quantitative estimate of the convergence rate in the first few Parareal iterations.enHyperbolic PDEParallel-in-time integrationPararealPseudo-spectral radiusPseudo-spectrumSpatial coarseningComputer Science, Information and General Works::004: Computer SciencesNatural Sciences and Mathematics::510: MathematicsConference Paper10.1007/978-3-032-17810-7_4