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

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.