A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations
2021; Cambridge University Press; Volume: 14; Issue: 4 Linguagem: Inglês
10.4208/nmtma.oa-2020-0020
ISSN2079-7338
AutoresXian‐Ming Gu, Bruno Carpentieri, Yongliang Zhao, Yu-Yun Huang, Xi-Le Zhao, Bruno Carpentieri, Yu-Yun Huang,
Tópico(s)Differential Equations and Numerical Methods
ResumoThe p-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci.Comput.,40(2)(2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108,2020, 17 pages]).However, when the BDFp (2 ≤ p ≤ 6) method is used to solve time-dependent PDEs, the generalization of these studies is not straightforward as p-step BDF is not selfstarting for p ≥ 2. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system.We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners.Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details.An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices.Finally, numerical experiments involving both the one-and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.
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