Multigrid Reduction in Time for non-linear hyperbolic equations

TL;DR

This paper investigates the performance of the Multigrid Reduction in Time (MGRIT) algorithm for non-linear hyperbolic PDEs, focusing on stability, convergence, and the impact of discretisation schemes like WENO and SSP.

Contribution

It provides a detailed analysis of MGRIT's effectiveness on hyperbolic equations and proposes techniques to enhance its convergence with low-order schemes.

Findings

CFL limit is the main factor affecting convergence speed.

High-accuracy WENO and SSP schemes influence MGRIT performance.

Performance degradation is linked to stability and discretisation choices.

Abstract

Time-parallel algorithms seek greater concurrency by decomposing the temporal domain of a Partial Differential Equation (PDE), providing possibilities for accelerating the computation of its solution. While parallelisation in time has allowed remarkable speed-ups in applications involving parabolic equations, its effectiveness in the hyperbolic framework remains debatable: growth of instabilities and slow convergence are both strong issues in this case, which often negate most of the advantages from time-parallelisation. Here, we focus on the Multigrid Reduction in Time (MGRIT) algorithm, investigating in detail its performance when applied to non-linear conservation laws with a variety of discretisation schemes. Specific attention is given to high-accuracy Weighted Essentially Non-Oscillatory (WENO) reconstructions, coupled with Strong Stability Preserving (SSP) integrators, which are often the discretisations of choice for such PDEs. A technique to improve the performance of MGRIT when applied to a low-order, more dissipative scheme is also outlined. This study aims at identifying the main causes for degradation in the convergence speed of the algorithm, and finds the Courant-Friedrichs-Lewy (CFL) limit to be the principal determining factor.