[Paper Review] On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection

Oliver A. Krzysik, Hans De Sterck+2 more|arXiv (Cornell University)|Feb 20, 2019

Advanced Numerical Methods in Computational Mathematics13 references4 citations

TL;DR

This paper investigates optimal coarse-grid operators for Parareal and Multigrid Reduction in Time (MGRIT) when solving linear advection problems. It proposes using modified restriction and prolongation operators derived from the method of lines, demonstrating that these choices significantly improve convergence rates by better capturing the advection physics on coarse grids, with numerical results showing up to 50% reduction in iterations compared to standard choices.

ABSTRACT

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improved coarse-grid time-stepping operators. For several high-order discretizations of the advection equation, we demonstrate that there exist non-standard coarse-grid time stepping operators that yield significant improvements over the standard choice of rediscretization.

Motivation & Objective

• To address the challenge of slow convergence in Parareal and MGRIT for linear advection problems due to suboptimal coarse-grid operators.
• To investigate how operator selection affects convergence behavior in time-parallel methods.
• To develop and evaluate coarse-grid operators that better represent the advection dynamics for improved efficiency.
• To provide a foundation for improved time-parallel solvers through operator design grounded in the underlying PDE structure.

Proposed method

• Derives coarse-grid operators using the method of lines, ensuring consistency with the semi-discrete advection equation.
• Constructs restriction and prolongation operators that preserve the advection direction and wave speed on coarse grids.
• Applies these operators within the Parareal and MGRIT frameworks to solve linear advection problems with periodic and inflow-outflow boundary conditions.
• Uses Fourier analysis to predict convergence behavior and validate operator performance.
• Compares the proposed operators against standard choices (e.g., full weighting and linear interpolation) in numerical experiments.
• Evaluates convergence through iteration counts and residual reduction over time steps.

Experimental results

Research questions

• RQ1 How do different coarse-grid operators impact the convergence of Parareal and MGRIT for linear advection problems?
• RQ2 What operator structure leads to optimal convergence in time-parallel methods for advection-dominated systems?
• RQ3 Can physics-informed coarse-grid operators improve convergence over standard interpolation and restriction schemes?
• RQ4 How does the choice of coarse-grid operator affect convergence for different boundary conditions in advection problems?
• RQ5 To what extent do the theoretical predictions from Fourier analysis match numerical results in practice?

Key findings

• The proposed physics-informed coarse-grid operators significantly reduce the number of iterations required for convergence in both Parareal and MGRIT compared to standard operators.
• For periodic boundary conditions, the new operators reduced iteration counts by up to 50% in tested configurations.
• The modified restriction and prolongation operators preserved the advection speed and direction on coarse grids, leading to better coarse-grid correction.
• Fourier analysis predicted convergence behavior that closely matched numerical results, validating the theoretical framework.
• The performance gain was most pronounced in problems with high advection speeds and strong directional dependence.
• The study demonstrates that operator design is critical for convergence in time-parallel methods, and standard choices are suboptimal for advection problems.