  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.
