The Parareal parallel-in-time integration method often performs poorly when applied to hyperbolic partial differential equations. This effect is even more pronounced when the coarse propagator uses a reduced spatial resolution. However, some combinations of spatial discretization and numerical time stepping nevertheless allow for Parareal to converge with monotonically decreasing errors. This raises the question how these configurations can be distinguished theoretically from those where the error initially increases, sometimes over many orders of magnitude. For linear problems, we prove a theorem that implies that the 2-norm of the Parareal iteration matrix is not a suitable tool to predict convergence for hyperbolic problems when spatial coarsening is used. We then show numerical results that suggest that the pseudo-spectral radius can reliably indicate if a given configuration of Parareal will show transient growth or monotonic convergence. For the studied examples, it also provides a good quantitative estimate of the convergence rate in the first few Parareal iterations.

翻译：Parareal并行时间积分方法在应用于双曲型偏微分方程时通常表现不佳。当粗粒度传播子采用降低的空间分辨率时，这种效应更为显著。然而，某些空间离散化与数值时间步进方法的组合仍能使Parareal实现误差单调递减的收敛。这引出了一个理论问题：如何从理论上区分这类配置与那些误差初始阶段会增长（有时达多个数量级）的配置。针对线性问题，我们证明了一个定理，表明当采用空间粗化时，Parareal迭代矩阵的2-范数不适用于预测双曲型问题的收敛行为。随后通过数值结果证明，伪谱半径能够可靠地指示给定Parareal配置是否会出现瞬态增长或单调收敛。对于所研究的算例，该指标还能为前几次Parareal迭代的收敛速率提供良好的定量估计。