Clusters worldwide have millions of cores. It is important to develop algorithms that can fully use those resources. Parallel-in-time algorithms parallelize simulations by taking advantage of additional cores to start computing the solution at a later time, even if the solution at the current time has not been computed yet.

We study three algorithms: one which is easy to use, Parareal, and a variant with more layers of computation, Multigrid Reduction-in-Time, as well as a more performant one, Space-Time Multigrid. We analyze their convergence properties to make them more efficient and to better understand them.