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

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

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