Parallel-in-time preconditioner for the Sinc-Nyström systems
Jun Liu, Shu‐Lin Wu · SIAM Journal on Scientific Computing · 2022
The sinc-Nyström method is a high-order numerical method based on sinc basis functions for discretizing evolutionary differential equations in time. But in this method we have to solve all the time steps in one shot (i.e., all-at-once), which results in a large-scale nonsymmetric dense system that is expensive to handle. In this paper, we propose and analyze a parallel-in-time preconditioner for such a dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and it has two advantages. First, we show that the eigenvalues of the preconditioned system are highly clustered with some uniform bounds which are independent of the mesh parameters. Second, the preconditioner can be computed in parallel for all the sinc time points via a block diagonalization procedure. Such a parallel potentials owing to the fact that the eigenvector matrix of the diagonalization is well conditioned. In particular, we show that the condition number of the eigenvector matrix only mildly grows as the number of sinc time points increases, and thus the roundoff error arising from the diagonalization procedure is controllable. The effectiveness of our proposed parallel-in-time preconditioners is verified by the observed mesh-independent convergence rates of the preconditioned GMRES in several numerical examples.