# Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

## TL;DR

This paper establishes a mathematical foundation for PFASST, a parallel-in-time method for linear PDEs, proving its convergence and analyzing its spectral properties in stiff and non-stiff regimes.

## Contribution

It provides the first rigorous convergence analysis of PFASST for linear problems using a multigrid framework and derives bounds on its spectral radius.

## Key findings

- PFASST is proven to be a convergent iterative method for linear problems.
- Spectral radius bounds are derived for both stiff and non-stiff cases.
- Results are validated through numerical experiments.

## Abstract

For time-dependent partial differential equations, parallel-in-time integration using the "parallel full approximation scheme in space and time" (PFASST) is a promising way to accelerate existing space-parallel approaches beyond their scaling limits. Inspired by the classical Parareal method and multigrid ideas, PFASST allows to integrate multiple time-steps simultaneously using a space-time hierarchy of spectral deferred correction sweeps. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. Very recently, however, PFASST for linear problems has been identified as multigrid method and in this paper, we will use this multigrid formulation and in particular PFASST's iteration matrix to show that in the non-stiff as well as in the stiff limit PFASST indeed is a convergent iterative method. We will provide upper bounds for the spectral radius of the iteration matrix and investigate how PFASST performs for increasing numbers of parallel time-steps. Finally, we will demonstrate that the results obtained here indeed relate to actual PFASST runs.