| Abstract: | The $p$-step backward difference formula (BDF) for solving systems ofODEs 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 Linand 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 $p$-stepBDF is not selfstarting for $p$ ≥ 2. In this note, we focus on the 2-step BDF which isoften 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-rankperturbation of a block triangular Toeplitz system. We first give an estimation of thecondition number of the all-at-once systems and then, capitalizing on previous work,we propose two block circulant (BC) preconditioners. Both the invertibility of thesetwo BC preconditioners and the eigenvalue distributions of preconditioned matricesare discussed in details. An efficient implementation of these BC preconditioners isalso presented, including the fast computation of dense structured Jacobi matrices.Finally, numerical experiments involving both the one- and two-dimensional Rieszfractional diffusion equations are reported to support our theoretical findings. |
