Interweaving PFASST and Parallel Multigrid

TL;DR

This paper explores integrating multigrid methods within PFASST to improve the efficiency of parallel time integration for PDEs, analyzing different coupling strategies and their impact on performance.

Contribution

It introduces and evaluates strategies for coupling PFASST with multigrid solvers, focusing on implicit diffusion treatment in PDEs, and compares full accuracy versus fixed V-cycle approaches.

Findings

Coupling PFASST with multigrid can enhance parallel efficiency.

Using fewer V-cycles reduces per-iteration cost but may increase total iterations.

Numerical experiments demonstrate trade-offs between accuracy and efficiency.

Abstract

The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple time-steps in parallel. However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any spatial system arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each sub-step and each iteration by some unspecified procedure. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach. This paper presents an initial investigation on the relative performance of different strategies for coupling PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each sub-step with a small fixed number of multigrid V-cycles. This reduces the cost of each PFASST iteration at the possible expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency of the resulting methods is explored through numerical examples.