Local Fourier Analysis of a Space-Time Multigrid Method for DG-SEM for the Linear Advection Equation

TL;DR

This paper analyzes the convergence of a space-time multigrid solver for hyperbolic PDEs using high-order discontinuous Galerkin methods, demonstrating effective convergence rates at high CFL numbers.

Contribution

It provides a local Fourier analysis of a novel space-time multigrid method with high-order DG-SEM discretizations, including coarsening strategies and convergence factors.

Findings

Convergence factors around 0.5 for first order, 0.375 for second order at high CFL.

Numerical experiments show even better rates around 0.3 and 0.25.

Effective multigrid convergence for high-order space-time discretizations.

Abstract

In this paper we present a Local Fourier Analysis of a space-time multigrid solver for a hyperbolic test problem. The space-time discretization is based on arbitrarily high order discontinuous Galerkin spectral element methods in time and a first order finite volume method in space. We apply a block Jacobi smoother and consider coarsening in space-time, as well as temporal coarsening only. Asymptotic convergence factors for the smoother and the two-grid method for both coarsening strategies are presented. For high CFL numbers, the convergence factors for both strategies are $0.5$ for first order, and $0.375$ for second order accurate temporal approximations. Numerical experiments in one and two spatial dimensions for space-time DG-SEM discretizations of varying order gives even better convergence rates of around $0.3$ and $0.25$ for sufficiently high CFL numbers.