Toward Parallel in Time for Chaotic Dynamical Systems

TL;DR

This paper introduces a modified multigrid in time method with a new coarsening scheme to improve parallel time-integration of chaotic systems, demonstrated on the Lorenz model.

Contribution

It proposes a novel time-coarsening scheme and modifications to nonlinear FAS multigrid to enhance convergence for chaotic initial value problems.

Findings

Significantly improved convergence of MGRIT for chaotic systems.

Effective capture of long-term behavior on coarse grids.

Numerical validation on the Lorenz system.

Abstract

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.