Appears in collection : Jean-Morlet Chair - PinT 2022: 11th Conference on Parallel-in-Time Integration / Chaire Jean-Morlet 2022 - 11ème conférence PinT - Parallel-in-Time Integration

The parallel-in-time integration of wave-type equations is well known to be a difficult task. When applying classical waveform-relaxation (WR) and parareal type methods, one generally experiences rapid error growth before reaching convergence in a finite number of iterations. This negative behavior prevents, in general, the successful application of these domain decomposition methods. In this talk, the focus is on WR-type methods. Classical WR convergence analyses use classical Laplace/Fourier techniques. However, these approaches provide analyses for unbounded t ime intervals, and do not allow one to describe precisely the WR converge behavior on finite t ime intervals. In this talk, we present a novel analysis based on the methods of characteristics, which allows us, on the one hand, to obtain a detailed characterization of the error growth along with the iterations and, on the other hand, to introduce a new parallel-in-time computational strategy. Numerical experiments support our new theoretical and numerical findings. This is a joint work with Martin J. Gander and Ilario Mazzieri.