A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

翻译：并行计算领域长期存在的问题是标准迭代并行时间方法对双曲型偏微分方程（PDE）以及对流主导型PDE的收敛性较差。本文针对多重网格时间约简（MGRIT）迭代并行时间方法的两层变体，推导了局部傅里叶分析（LFA）收敛理论。该闭合形式理论揭示了采用粗网格对细网格问题重新离散化的标准方法时，MGRIT在对流主导型PDE上收敛性差的根源。具体表现为：收敛性差至少部分源于对某些光滑傅里叶模式（即特征分量）的粗网格校正不足——此前研究已指出这正是稳态对流主导型PDE的经典空间多重网格收敛性差的原因。应用该收敛理论，我们证明对于某些半拉格朗日离散化的对流问题，采用重新离散化粗网格算子的MGRIT收敛性无法对CFL数或粗化因子保持鲁棒性。这一分析的重要启示在于：空间多重网格中改善收敛性的技术可被改造应用于MGRIT框架，从而开发更鲁棒的并行时间求解器。该策略已在近期工作中取得显著成效，而本文则进一步从理论层面验证了该方法的有效性。