Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs), but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.

翻译：许多迭代平行算法已证明对于以扩散为主的部分差异方程式(PDEs)非常高效,但在应用到以对映为主的PDEs时效率低甚至差异很大。我们考虑将多格即时递减算法(MGRIT)算法应用于线性对映式 PDEs。与这种方法高效时间整合的关键是使用一个粗格电网操作员,该操作员对所谓的理想粗格电网操作员提供足够准确的近似。对于某些半拉格朗差分解型,我们展示了一个新的半拉格朗加半共电网操作员,该操作员可以快速和可伸缩的多时间整合线性PDEs。粗格电网操作员由半拉格递减算法的离解算法组成,随后是修正术语,因此,复合操作员的领先顺序调整差差差差差差差差差差差差差差差差差差差差差差差差差差差差数与理想的半拉格网格操作员的操作员差差差差差分解分解分解。平行结果显示,在可变波速度的半拉格网格网格操作员的半加网内操作员基网内操作,从而导致线对线式对线线的多速度的多调和宽差差差差差差差差差差差差差差差差差差差差差差的多的多时间集问题,用。使用一个和多的计算法在一个和差法的计算法,用法,用法在一个和分法上,用法则用法则用法制后用法,用法,用法将一个和分法将一个和两个制成,在一个和两个的分算法上,用上,用上,用上一个和两个不同的分法上一个分算法上一个直达法上,用办法提供一个和两种办法提供一种,用上一个和两个上一个和两个上一个和分算法制办法提供一种办法提供一个和两个的分差差差差差差差差差差差差法,用办法,用。