For a given temporal discretization scheme, the iterative diagonalization-based parallel-in-time (ParaDiag) algorithm has shown excellent performance in accelerating computations for evolutionary problems, particularly for hyperbolic equations. Recently, considerable efforts have been made towards the spectral analysis of the involved iteration matrix \({\mathscr {M}}\) . Numerous theoretical results suggest that if the numerical discretization scheme is stable, then the iteration matrix \({\mathscr {M}}\) has a good spectral clustering, thereby enabling fast convergence of the iterative ParaDiag algorithm. However, if the numerical scheme is unstable, no clear result is obtained yet, although extensive numerical experiments indicate that unstable scheme generally leads to divergence of the iterative ParaDiag. To clarify this issue, in this paper we investigate two parameterized Numerov-type methods, namely Hairer’s and Chawla’s methods, which are employed as the temporal discretization schemes for solving the wave equation. We demonstrate that the stability of the numerical scheme is not a necessary condition for rapid convergence of the iterative Paradiag algorithm. Even when the stability conditions of Hairer’s and Chawla’s methods are not satisfied, the spectrum of the iteration matrix \({\mathscr {M}}\) still exhibits good clustering behavior, enabling the iterative ParaDiag algorithm to converge rapidly, however, to an unstable solution of the numerical scheme.
