Stability implies robust convergence of a class of diagonalization-based iterative algorithms

Solving wave equations in a time-parallel manner is challenging, and the algorithm based on the block α-circulant preconditioning technique has shown its advantage in many existing studies (where α ∈ (0, 1) is a free parameter). Considerable efforts have been devoted to exploring the spectral radius of the preconditioned matrix and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified way to analyze the convergence via directly studying the error of the algorithm and using the stability of the time integrator. Our analysis works for all one-step time-integrators and two exemplary classes of two-step time-integrators: the parameterized Numerov methods and the parameterized two-stage hybrid methods. The main conclusion is that the global error satisfies ‖err‖W ,2 ≤ α 1−α ‖err‖W ,2 provided that the time-integrator is stable, where k is the iteration index and for any vector v the norm ‖v‖W ,2 is defined by ‖v‖W ,2 = ‖W v‖2 with W being a matrix depending on the space discretization matrix only. Even though we focus on wave equations in this paper, the proposed argument is directly applicable to other evolution problems.