  McDonald, Pestana and Wathen (SIAM J. Sci. Comput. 40(2), pp. A2012-A1033,
2018) present a method for preconditioning of time-dependent PDEs via
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. Our simple implementation (in C++ and MPI) is available at
the Git repository PARALAAOMPI.
https://github.com/anthonyjamesgoddard/PARALAAOMPI
