The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the 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, 17 pages]. However, these studies ignored that the $p$-step BDF ($p\geq 2$) is not selfstarting, when they are exploited to solve time-dependent PDEs. 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, but its resultant all-at-once discretized system is a block triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first give an estimation of the condition number of the all-at-once systems and then adapt the previous work to construct two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. The efficient implementation of these BC preconditioners is also presented especially for handling the 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.
翻译:用于解决 ODE 系统的美元向后退差异公式(BDF ) 。 然而,这些研究忽视了一种全线性系统(BDF $p$-sep BDF ($p\geq 2$) 并不是自动启动的,而这种系统是用来解决基于时间的Krylov 子空间解决方案(见麦当劳、佩斯塔纳和Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033) 和 Lin 和 Ng [arXiv:2002.01108, 17页] 。然而,这些研究忽视了这种系统不是自动启动的,而是用来解决依赖时间的 Prylov 子空间解决方案的全线性系统。 在本说明中,我们关注通常优于解决 Riesz 分解分解公式的陷阱规则的双步BDFFFDFS 。 但其全自动分解系统是一个三角系统, 并且是一个低级的 。我们首先估算了全部的 BDFDF 系统的条件号系统,然后又将两个预估值的 IMexreal- dealrealrealderealrealreal realrealrealrealrealreailveilveal 。