Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes known as characteristic components.We propose an alternative coarse-grid that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate substantial speed-up over sequential time-stepping.

翻译：偏微分方程（PDE）的并行时域方法近年来发展迅速，尤其在扩散主导问题中效果显著。然而，文献广泛指出，许多此类方法在对流主导问题中表现较差。本文针对恒定波速线性对流问题的离散格式，分析了特定的迭代并行时域算法——多网格时域约化（MGRIT）方法。我们重点关注采用空间迎风有限差分与时间龙格-库塔方法的常见线法离散格式。利用先前研究中的收敛性框架，我们证明对于这些离散格式的一个子类，若采用粗网格上对细网格问题重新离散的标准方法，则无法实现MGRIT相对于CFL数和粗化因子的稳健收敛。这种收敛性差和非稳健性部分源于对光滑傅里叶模式（即特征分量）的粗网格校正不充分。我们提出一种替代粗网格方案，可对此类模式提供更优的校正。该粗网格算子与先前工作相关，采用半拉格朗日离散结合隐式处理的截断误差校正。理论与数值实验表明，该粗网格算子使得MGRIT在所考虑的多种线法离散格式（包括高阶隐式和显式离散）中均能实现快速收敛。并行结果显示出相较于顺序时间推进的显著加速效果。