Volume 51, pp. 135-150, 2019.

A note on parallel preconditioning for all-at-once evolutionary PDEs

Anthony Goddard and Andy Wathen


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.

Full Text (PDF) [383 KB], BibTeX

Key words

parallel-in-time, monolithic method, preconditioning

AMS subject classifications

65M20, 65F08, 65Y05

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