On Fourier analysis of polynomial multigrid for arbitrary multi-stage cycles

TL;DR

This paper uses Fourier analysis to study polynomial multigrid methods with arbitrary multi-stage cycles, revealing how cycle asymmetry and smoothing strategies improve convergence in solving advection-diffusion and Navier-Stokes equations.

Contribution

It provides a novel Fourier analysis framework for polynomial multigrid with arbitrary cycles, demonstrating how cycle asymmetry enhances convergence in dual-time schemes.

Findings

V-cycle asymmetry improves convergence

Additional prolongation smoothing is beneficial

Analytic results are confirmed numerically in Navier-Stokes simulations

Abstract

The Fourier analysis of the \emph{p}-multigrid acceleration technique is considered for a dual-time scheme applied to the advection-diffusion equation with various cycle configurations. It is found that improved convergence can be achieved through \emph{V}-cycle asymmetry where additional prolongation smoothing is applied. Experiments conducted on the artificial compressibility formulation of the Navier--Stokes equations found that these analytic findings could be observed numerically in the pressure residual, whereas velocity terms---which are more hyperbolic in character---benefited primarily from increased pseudo-time steps.