A note on parallel preconditioning for all-at-once evolutionary PDEs. ETNA - Electronic Transactions on Numerical Analysis

McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40 (2018), pp. A1012–A1033] present a method for preconditioning time-dependent PDEs via an approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach which also allows for efficient parallel execution. Results are shown for both parabolic and hyperbolic PDEs. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI