| Abstract: | The $p$-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 et al. 36] and Lin
and Ng 32]). However, when the BDF$p$ (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. |
