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Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs

Gobinda Garai, Nagaiah Chamakuri

TL;DR

A novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework, and is generalized to nonlinear problems, for which convergence is rigorously demonstrated.

Abstract

This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential integrators. We establish the convergence of the proposed methods both as preconditioned fixed-point iterations and as precon- ditioners within the GMRES framework. Furthermore, we extend the Exp-ParaDiag formulation to achieve sixth-order temporal accuracy using exponential integrators. The proposed approach is also generalized to nonlinear problems, for which convergence is rigorously demonstrated. A series of numerical experiments is presented to validate the theoretical results and to illustrate the robustness and efficiency of the developed methods.

Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs

TL;DR

A novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework, and is generalized to nonlinear problems, for which convergence is rigorously demonstrated.

Abstract

This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential integrators. We establish the convergence of the proposed methods both as preconditioned fixed-point iterations and as precon- ditioners within the GMRES framework. Furthermore, we extend the Exp-ParaDiag formulation to achieve sixth-order temporal accuracy using exponential integrators. The proposed approach is also generalized to nonlinear problems, for which convergence is rigorously demonstrated. A series of numerical experiments is presented to validate the theoretical results and to illustrate the robustness and efficiency of the developed methods.
Paper Structure (22 sections, 23 theorems, 80 equations, 22 figures, 2 tables)

This paper contains 22 sections, 23 theorems, 80 equations, 22 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulation and Convergence of Exp-ParaDiag
  3. Convergence in a discrete setting
  4. Optimization of the free parameter $\alpha$
  5. Preconditioner for $\mathcal{O}(\Delta t)$ Scheme
  6. Eigenvalue estimate of the preconditioned system
  7. Convergence Analysis in GMRES Setting
  8. Preconditioner for $\mathcal{O}((\Delta t)^2)$ Scheme
  9. Eigenvalue estimate of the preconditioned system
  10. Analysis of the preconditioned system in GMRES setting
  11. Exp-ParaDiag Formulation for $\mathcal{O}((\Delta t)^s)$ Scheme for $3\leq s\leq 6$
  12. Formulation and Convergence in the Nonlinear Setting
  13. Numerical Illustration
  14. Experiment in Linear Settings
  15. Experiment of Exp-ParaDiag with $\mathcal{O}(\Delta t)$ Scheme
  16. ...and 7 more sections

Key Result

Theorem 2.1

The discrete error $\mathbf{e}_n^k$ of the Exp-ParaDiag method satisfies the following error contraction relation

Figures (22)

  • Figure 1: First: error comparison for $(a, c)=(0.1, 0)$ ; Second: error comparison for $(a, c)=(10^{-5}, 1)$; Third: 2nd order accurate Padé approximation; Fourth: 3rd order accurate Padé approximation.
  • Figure 2: First: Mesh refinement for $(a, c)=(0.01, 0.1)$ and $T=4$; Second: Mesh refinement for $(a, c)=(0.00001, 0.1)$ and $T=4$; Third: Long-time simulation for $(a, c)=(0.1, 0.1)$; Fourth: Long-time simulation for $(a, c)=(0.00001, 0.1)$.
  • Figure 3: First: convergence for $(a, c)=(0.1, 0.1)$; Second: convergence for $(a, c)=(0.00001, 0.1)$; Third: Spectrum of $Q_{\alpha}^{-1}Q$ and its bound for $(a, c)=(0.1, 0.1)$; Fourth: Spectrum of $Q_{\alpha}^{-1}Q$ and its bound for $(a, c)=(0.00001, 0.1)$.
  • Figure 4: First: convergence for various $T$ with $(a, c)=(0.1, 0.1)$; Second: convergence for various $T$ with $(a, c)=(0.00001, 0.1)$; Third: convergence for various $\alpha$ with $(a, c)=(0.1, 0.1)$; Fourth: convergence for various $\alpha$ with $(a, c)=(0.00001, 0)$.
  • Figure 5: First: mesh independence with $(a, c)=(0.1, 0.1)$; Second: mesh independence with $(a, c)=(0.00001, 0.1)$; Third: convergence for different $T$; Fourth: real part of the solution to the SE.
  • ...and 17 more figures

Theorems & Definitions (44)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 34 more