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A Parareal algorithm without Coarse Propagator?

Martin J. Gander, Mario Ohlberger, Stephan Rave

TL;DR

This paper investigates a Parareal time-parallel algorithm variant that intentionally omits the coarse propagator $G$, focusing on parabolic problems such as the 1D heat equation. By employing a spectral discretization in space, the authors show that Parareal updates diagonalize in Fourier modes, yielding an explicit convergence bound $\sup_n \|U_n^k(\cdot)-u(\cdot,T_n)\|_2 \le e^{-(m_G+1)^2 k\Delta T} \sup_n \|U_n^0(\cdot)-u(\cdot,T_n)\|_2$, which remains valid even when $m_G=0$ (no coarse propagator). Numerical experiments reveal that, under Dirichlet boundaries, the method converges without $G$ and can even converge faster for larger time-step slices, whereas under Neumann boundaries convergence may fail unless very few coarse intervals are used. The work highlights a sharp boundary-condition dependence and proves a scalability result (Theorem TH1) for fixed time-step length, while noting that hyperbolic problems do not enjoy the same forgetting properties and require coarse corrections for robust convergence. Overall, the results advance understanding of time-parallelization in parabolic PDEs and delineate the limitations when extending to hyperbolic dynamics.

Abstract

The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.

A Parareal algorithm without Coarse Propagator?

TL;DR

This paper investigates a Parareal time-parallel algorithm variant that intentionally omits the coarse propagator , focusing on parabolic problems such as the 1D heat equation. By employing a spectral discretization in space, the authors show that Parareal updates diagonalize in Fourier modes, yielding an explicit convergence bound , which remains valid even when (no coarse propagator). Numerical experiments reveal that, under Dirichlet boundaries, the method converges without and can even converge faster for larger time-step slices, whereas under Neumann boundaries convergence may fail unless very few coarse intervals are used. The work highlights a sharp boundary-condition dependence and proves a scalability result (Theorem TH1) for fixed time-step length, while noting that hyperbolic problems do not enjoy the same forgetting properties and require coarse corrections for robust convergence. Overall, the results advance understanding of time-parallelization in parabolic PDEs and delineate the limitations when extending to hyperbolic dynamics.

Abstract

The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.
Paper Structure (5 sections, 1 theorem, 18 equations, 4 figures)

This paper contains 5 sections, 1 theorem, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A Numerical Experiment
  3. Convergence Analysis
  4. Conclusions
  5. Acknowledgement.

Key Result

Theorem 1

The Parareal algorithm Parareal for the heat equation HeatEquation1D using the spectral coarse propagator CoarseApproximation and the spectral fine propagator FineApproximation satisfies for any initial guess $U_n^0$ the convergence estimate and this estimate also holds if the coarse propagator does not contain any modes, $m_G=0$, which means it is not present. The Parareal algorithm therefore co

Figures (4)

  • Figure 1: Running a classical Parareal algorithm with and without coarse propagator. From top left to bottom right: 48, 24, 12, and 6 coarse time intervals with $T=3$.
  • Figure 2: Classical Parareal with and without coarse propagator with zero Neumann boundary conditions. From top left to bottom right: 48, 24, 12, and 6 coarse time intervals and $T=3$.
  • Figure 3: From left to right: solution of the heat equation model problem with zero Dirichlet and zero Neumann boundary conditions. Solution of the corresponding advection equation with Dirichlet boundary condition on the left and periodic boundary condition, and solution of the second order wave equation with zero Dirichlet boundary conditions.
  • Figure 4: Contraction for each Fourier mode $m$ as function of the Parareal time interval length $\Delta T$ with (blue, PdT=$\Delta T$) and without (green, PNCdT=$\Delta T$) coarse propagator.

Theorems & Definitions (5)

  • Theorem 1: Parareal convergenve even without coarse propagator
  • proof
  • Remark 1
  • Remark 2
  • Remark 3