  As CPU clock speeds have stagnated, and high performance computers continue
to have ever higher core counts, increased parallelism is needed to take
advantage of these new architectures. Traditional serial time-marching schemes
are a significant bottleneck, as many types of simulations require large
numbers of time-steps which must be computed sequentially. Parallel in Time
schemes, such as the Multigrid Reduction in Time (MGRIT) method, remedy this by
parallelizing across time-steps, and have shown promising results for parabolic
problems. However, chaotic problems have proved more difficult, since chaotic
initial value problems are inherently ill-conditioned. MGRIT relies on a
hierarchy of successively coarser time-grids to iteratively correct the
solution on the finest time-grid, but due to the nature of chaotic systems,
subtle inaccuracies on the coarser levels can lead to poor coarse-grid
corrections. Here we propose a modification to nonlinear FAS multigrid, as well
as a novel time-coarsening scheme, which together better capture long term
behavior on coarse grids and greatly improve convergence of MGRIT for chaotic
initial value problems. We provide supporting numerical results for the Lorenz
system model problem.
