Impact of spatial coarsening on Parareal convergence for the linear advection equation

TL;DR

This paper investigates why Parareal can exhibit strong transient growth when coupled with spatial coarsening in hyperbolic problems like linear advection, and why the conventional 2-norm of the iteration matrix fails as a predictor of convergence. Focusing on linear problems with normal system matrices, the authors prove a lower bound on the Parareal error-propagation norm that depends only on the fine propagator and the eigenstructure, showing that the 2-norm is not a reliable indicator of convergence. They introduce the ε-pseudo-spectrum and the associated pseudo-spectral radius ρ_ε(E) as effective tools to diagnose and quantify initial transient growth and to estimate early convergence rates, supported by numerical experiments on four Parareal configurations with varying spatial discretizations and diffusion properties. The results demonstrate that the pseudo-spectrum can reliably distinguish configurations with monotone convergence from those with transient growth, suggesting a practical criterion for designing coarse propagators. The study highlights the potential of pseudo-spectral analysis to guide the development of efficient parallel-in-time schemes for hyperbolic problems and indicates directions for optimizing coarse propagators while providing reproducible code for further exploration.

Abstract

The Parareal parallel-in-time integration method often performs poorly when applied to hyperbolic partial differential equations. This effect is even more pronounced when the coarse propagator uses a reduced spatial resolution. However, some combinations of spatial discretization and numerical time stepping nevertheless allow for Parareal to converge with monotonically decreasing errors. This raises the question how these configurations can be distinguished theoretically from those where the error initially increases, sometimes over many orders of magnitude. For linear problems, we prove a theorem that implies that the 2-norm of the Parareal iteration matrix is not a suitable tool to predict convergence for hyperbolic problems when spatial coarsening is used. We then show numerical results that suggest that the pseudo-spectral radius can reliably indicate if a given configuration of Parareal will show transient growth or monotonic convergence. For the studied examples, it also provides a good quantitative estimate of the convergence rate in the first few Parareal iterations.

Figures (7)

• Figure 1: Numerical solution (blue) generated by the four configurations shown in Table \ref{['tab:configs']} after $P-1 = 9$ Parareal iterations and analytic solution (red) for $u_0(x) = \sin(2 \pi x) + \sin(8 \pi x)$ (upper four) and $u_0(x) = H(x-0.5)$ with $H$ being the Heaviside function (lower four). However, note that the Parareal iteration matrix defined and studied later does not depend on the initial value. The purpose of this figure is only to visualize the numerical properties of the four configurations.
• Figure 2: Blue circles show eigenvalues of $A$ for a first order upwind (upper and lower left) and second order centered (upper and lower right) spatial finite difference discretisation of the linear advection equation. Red squares show eigenvalues of the exact time propagation operator (upper) and implicit Euler propagator (lower). Blue dots to the left of the imaginary axis indicate spatial numerical diffusion. Red squares inside the unit circle indicate numerical diffusion. The upper left configuration has spatial diffusion, the lower left spatial and temporal diffusion, the lower right has only temporal diffusion while the upper right has no diffusion.
• Figure 3: Norm of the Parareal iteration matrix $\left\| \mathbf{E} \right\|$ for changing fine- and coarse time step size $\Delta t = \delta t$. Configuration A but with changing coarse spatial resolution (upper left), configuration B with changing coarse resolution (upper right), a combination of centered finite difference and implicit Euler propagator and thus temporal but no spatial numerical diffusion (lower left) and configurations C/D with changing coarse spatial resolution (lower right). Note the changing scaling of the y-axes.
• Figure 4: Pseudo-spectrum of the Parareal iteration matrix for configurations A (upper left), B (upper right), C (lower left) and D (lower right). In all cases, $\left\| \mathbf{E} \right\|_2 > 1$. If the pseudo-spectrum is close to circles $B_{\varepsilon}$ of radius $\varepsilon$, we expect monotonic convergence of $\mathbf{E}^k$. By contrast, a substantially distorted pseudo-spectrum indicates initial transient growth and non-monotonic convergence of Parareal. The dashed line shows the unit circle for reference. For the pseudo-spectra without protrusions (upper left and lower right) we see monotonic convergence.
• Figure 5: Convergence of the four different Parareal configurations from Table \ref{['tab:configs']} against the fine solution. Upper left: configuration A, upper right: configuration B, lower left: configuration C and lower right: configuration D. While the norm of the Parareal iteration matrix $\left\| \mathbf{E} \right\|_2$ is larger than unity for all configurations, the pseudo-spectral radius $\rho_{\varepsilon}(\mathbf{E})$ with $\varepsilon=0.1$ correctly distinguishes between initial increase of $\left\| \mathbf{e}^k \right\|_2$ (upper right and lower left) and monotonic decrease (upper left and lower right). It also gives a good quantitative prediction of the early convergence rate of Parareal.

Theorems & Definitions (20)

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