An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

TL;DR

This paper compares two parallel-in-time numerical methods, PFASST and space-time multigrid, for solving reaction-diffusion problems, analyzing their scalability, convergence, and performance on linear and nonlinear cases.

Contribution

It provides a detailed experimental comparison of PFASST and space-time multigrid methods, highlighting their differences in convergence and scalability for reaction-diffusion equations.

Findings

Both methods achieve similar discrete solutions at time nodes.

Strong and weak scaling behaviors differ between methods.

Convergence behavior varies for nonlinear problems.

Abstract

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time-dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.