Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

TL;DR

It is shown that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor, and techniques developed for improving convergence in the spatial multigrid context can be re-purposed to develop more robust parallel-in-time solvers.

Abstract

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

Figures (3)

• Figure 1: Plots in Fourier frequency space of quantities relating to the LFA spectral radius in \ref{['lem:rho_E_space-time']} for a $p = 3$ semi-Lagrangian discretization of linear advection using rediscretization on the coarse grid. The coarsening factor is $m = 4$, FCF-relaxation is used, and the CFL number is $c = 0.8$. Top left: Difference between symbols of coarse-grid operator and ideal coarse-grid operator. Top right: Symbol of coarse-grid operator. Bottom left: Contour of spectral radius evaluated with $1024$ points in $\omega$ and $1024/m$ points in $\theta$, with its maxima over the whole space marked with magenta diamonds. Bottom right: Cross-sections of the spectral radius along the green and blue lines pictured in the bottom left panel. Dashed green lines mark characteristic components with frequency $\theta(\omega) = - \frac{\omega \alpha \delta t}{h}$, while blue lines mark the slowest converging modes $\theta(\omega) = \theta^{\dagger}(\omega) = \frac{1}{m} \arg \mu(\omega)$ (see \ref{['thm:Ei_norm']}).
• Figure 2: Numerical confirmation of \ref{['THM:RHO-LWR-BOUND-SL']}: Convergence factors for the semi-Lagrangian discretizations $\Phi = {\cal S}_p^{(\delta t)}$, with $p = 1$ (left), and $p = 3$ (right), as a function of fractional fine-grid CFL number $\varepsilon^{(\delta t)}$ when rediscretizing on the coarse grid, $\Psi = {\cal S}_p^{(m \delta t)}$. MGRIT uses FCF-relaxation ($\nu = 1$), and a coarsening factor of $m$. Thin solid lines are the LFA convergence factor $\max \limits_{(\omega, \theta) \in [-\pi, \pi) \times \Theta^{\rm low}} \rho( \widehat{{\cal E}}(\omega, \theta) )$ obtained by discretizing $\omega$ with $128$ points. Thick dashed lines are the function $\widecheck{\rho}_p ( \varepsilon^{(\delta t)} )$ from \ref{['eq:rho-check']} that acts as a lower bound on the LFA convergence factor (see \ref{['eq:rho-SL-lwr-bnd']}). Filled triangle markers are $\varepsilon^{(\delta t)} = 1 - \frac{2}{3 m}$, the right-hand end point of the interval in \ref{['eq:rho-check-interval']} on which $\widecheck{\rho}_p ( \varepsilon^{(\delta t)} )$ is shown to exceed unity. Open circle markers are experimentally measured effective convergence factors of MGRIT on a finite interval $t \in (0, T]$ taken from Figure 2 of De Sterck et al.DeSterck_etal_2023_SL
• Figure 3: Examples of the functions analyzed in \ref{['app:conv-fac-constant']} for $p = 3$, with $m = 2$ (left), and $m = 6$ (right).

Theorems & Definitions (43)

• remark 1: Unit eigenvalues
• definition 1: $m \delta t$-harmonics
• lemma 1: F-relaxation Fourier symbol
• proof
• corollary 1: Idempotence of F-relaxation
• proof
• lemma 2: Pre-relaxation Fourier symbol
• proof
• theorem 1: Error propagator Fourier symbol
• proof