2021/08/03 by Jun Liu, Shu‐Lin Wu, Liu, Jun +1
Computer Science · Engineering · Mathematics · #65M70 #65Y05 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2108.01700
openalex publication_date 2021/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 preconditioner for such dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and 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 used parallel for all the Sinc time points via a block diagonalization procedure. Such a parallel potential owes 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 PinT preconditioners is verified by the observed mesh-independent convergence rates of the preconditioned GMRES in reported numerical examples.