Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A model order reduction based adaptive parareal method for time-dependent partial differential equations

Xiaoying Dai, Miao Hu, Shuwei Shen

TL;DR

This paper addresses the challenge of achieving scalable time-parallel solutions for time-dependent PDEs by coupling parareal with model order reduction. The core idea is to build an adaptive coarse propagator using POD-based reduced spaces derived from snapshots produced by the fine propagator, and to update the POD subspace in each parareal iteration, augmenting it with current initial values to reduce projection error. A thorough speed-up analysis and detailed numerical experiments on 3D advection–diffusion problems with Kolmogorov and ABC flows demonstrate that the adaptive method yields much higher accuracy than plain Parareal for long-time simulations, particularly in advection-dominated regimes. The work provides a practical framework for scalable space-time parallel computation and suggests directions for improving coarse propagator design and updating strategies in future research.

Abstract

In this paper, we propose a model order reduction based adaptive parareal method for time-dependent partial differential equations. By using the data obtained by the fine propagator in each iteration of the plain parareal method together with some model order reduction technique, we construct the coarse propagator adaptively in each parareal iteration, and then obtain our adaptive parareal method. We apply this new method to solve some 3D time-dependent advection-diffusion equations with the Kolmogorov flow and the ABC flow. Numerical results show the good performance of our method in simulating long-term evolution problems.

A model order reduction based adaptive parareal method for time-dependent partial differential equations

TL;DR

This paper addresses the challenge of achieving scalable time-parallel solutions for time-dependent PDEs by coupling parareal with model order reduction. The core idea is to build an adaptive coarse propagator using POD-based reduced spaces derived from snapshots produced by the fine propagator, and to update the POD subspace in each parareal iteration, augmenting it with current initial values to reduce projection error. A thorough speed-up analysis and detailed numerical experiments on 3D advection–diffusion problems with Kolmogorov and ABC flows demonstrate that the adaptive method yields much higher accuracy than plain Parareal for long-time simulations, particularly in advection-dominated regimes. The work provides a practical framework for scalable space-time parallel computation and suggests directions for improving coarse propagator design and updating strategies in future research.

Abstract

In this paper, we propose a model order reduction based adaptive parareal method for time-dependent partial differential equations. By using the data obtained by the fine propagator in each iteration of the plain parareal method together with some model order reduction technique, we construct the coarse propagator adaptively in each parareal iteration, and then obtain our adaptive parareal method. We apply this new method to solve some 3D time-dependent advection-diffusion equations with the Kolmogorov flow and the ABC flow. Numerical results show the good performance of our method in simulating long-term evolution problems.
Paper Structure (10 sections, 44 equations, 16 figures, 1 algorithm)

This paper contains 10 sections, 44 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adaptive parareal method based on model order reduction
  4. Spatial discretization of the fine propagator
  5. Adaptive spatial discretization of the coarse propagator
  6. Speed-up
  7. Numerical experiments
  8. Kolmogorov flow
  9. Arnold-Belt-Childress (ABC) flow
  10. Concluding remarks

Figures (16)

  • Figure 1: The evolution curves of the relative error of $U^k_n$ obtained by the method "Parareal" in each parareal iteration for solving the Kolmogorov flow with $\epsilon=0.5,0.1$
  • Figure 2: The evolution curves of the relative error of $U^k_n$ obtained by the method "AdapParareal" in each parareal iteration for solving the Kolmogorov flow with $(m_l,p)=(0,0)$ and $\epsilon=0.5,0.1$
  • Figure 3: The evolution curves of the relative error of $U^k_n$ obtained by the method "AdapParareal" in each parareal iteration for solving the Kolmogorov flow with $(m_l,p)=(1,0)$ and $\epsilon=0.5,0.1$
  • Figure 4: The evolution curves of the relative error of $U^k_n$ obtained by the method "AdapParareal" in each parareal iteration for solving the Kolmogorov flow with $(m_l,p)=(0,1)$ and $\epsilon=0.5,0.1$
  • Figure 5: The evolution curves of the relative error of $U^k_n$ obtained by the method "AdapParareal" in each parareal iteration for solving the Kolmogorov flow with $(m_l,p)=(1,1)$ and $\epsilon=0.5,0.1$
  • ...and 11 more figures

Theorems & Definitions (1)

  • Remark 1