Journal of Scientific Computing

, Volume 67, Issue 2, pp 644–668

# Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues

• Shu-Lin Wu
Article

## Abstract

Parareal is an iterative algorithm and is characterized by two propagators $$\mathscr {G}$$ and $$\mathscr {F}$$, which are respectively associated with large step size $$\varDelta T$$ and small step size $$\varDelta t$$, where $$\varDelta T=J\varDelta t$$ and $$J\ge 2$$ is an integer. The choice $$\mathscr {G}=\mathscr {F}=$$Backward-Euler denotes the simplest implicit parareal solver, which we call Parareal-Euler, and has been studied widely in recent years. For linear problem $$\mathbf {U}'(t)+\mathbf {A}\mathbf {U}(t)=\mathbf {g}(t)$$ with $$\mathbf {A}$$ being a symmetric positive definite matrix, this algorithm converges very fast and the convergence rate is insensitive to the change of J and $$\varDelta t$$. However, for the case that the spectrum of $$\mathbf {A}$$ contains complex values, no provable results are available in the literature so far. Previous studies based on numerical plotting show that we can not expect convergence for the Parareal-Euler algorithm on the whole right-hand side of the complex plane. Here, we consider a representative situation: $$\sigma (\mathbf {A})\subseteq \mathbf {D}(\theta ):=\left\{ (x,iy)\in \mathbf {C}: x\ge 0, |y|\le \tan (\theta )x\right\}$$ with $$\theta \in (0, \frac{\pi }{2})$$, i.e., the spectrum $$\sigma (A)$$ is contained in a sectorial region. Spectrum distribution of this type arises naturally for semi-discretizing a wide rang of time-dependent PDEs, e.g., the Fokker-Planck equations. We derive condition, which is independent of J and depends on $$\theta$$ only, to ensure convergence of the Parareal-Euler algorithm. Numerical results for initial value and time-periodic problems are provided to support our theoretical conclusions.

## Keywords

Parareal algorithm Backward-Euler Convergence analysis Complex eigenvalues

## Mathematics Subject Classification

65R20 45L05 65L20

## Notes

### Acknowledgments

The authors are very grateful to the anonymous referees for the careful reading of a preliminary version of the manuscript and their valuable suggestions and comments, which greatly improved the quality of this paper. This work was supported by the NSF of China (11301362, 11371157, 91130003), the NSF of Technology & Education of Sichuan Province (2014JQ0035,15ZA0220), the project of key laboratory of bridge non-destruction detecting and computing (2013QZY01) and the NSF of SUSE (2015LX01).

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