A note on parallel preconditioning for all-at-once evolutionary PDEs. 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.
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References in zbMATH (referenced in 9 articles , 1 standard article )
Showing results 1 to 9 of 9.
- Danieli, Federico; Southworth, Ben S.; Wathen, Andrew J.: Space-time block preconditioning for incompressible flow (2022)
- Sun, Yafei; Wu, Shu-Lin; Xu, Yingxiang: A parallel-in-time implementation of the Numerov method for wave equations (2022)
- Lin, Xue-lei; Ng, Michael: An all-At-once preconditioner for evolutionary partial differential equations (2021)
- Wu, Shuonan; Zhou, Zhi: A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations (2021)
- Gu, Xian-Ming; Wu, Shu-Lin: A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel (2020)
- Liu, Jun; Wu, Shu-Lin: A fast block (\alpha)-circulant preconditoner for all-at-once systems from wave equations (2020)
- Wu, Shu-Lin; Liu, Jun: A parallel-in-time block-circulant preconditioner for optimal control of wave equations (2020)
- Southworth, Ben S.: Necessary conditions and tight two-level convergence bounds for parareal and multigrid reduction in time (2019)
- Wathen, Anthony Goddard Andy: A note on parallel preconditioning for all-at-once evolutionary PDEs (2019)