论文标题
通过修改的半拉格朗日粗网格操作员，快速减少乘坐时间降低时间
Fast multigrid reduction-in-time for advection via modified semi-Lagrangian coarse-grid operators
论文作者
论文摘要
许多迭代平行算法已被证明对扩散为主导的偏微分方程（PDE）高效，但是当应用于对流为主的PDE时，效率低下甚至发散。我们考虑将多族降低时间（MGRIT）算法应用于线性对流PDE。与此方法有效整合的有效时间集成的关键是使用一个粗网格操作员，该操作员为所谓的理想的粗网格操作员提供了足够准确的近似值。对于某些半拉格朗日离散的类别，我们提出了一种新型的半拉格朗日基于的粗网格操作员，可导致线性对流PDE的快速且可扩展的多级时间集成。粗网格操作员由半拉格朗日离散化，然后是校正项组成，其校正设计使复合算子的前阶截断误差大致相当于理想的粗网格操作员的前阶截断误差。并行结果表明，在一个和两个空间维度中，对于可变速度速度的对流问题的顺序时间集成而言，大幅加速，并使用高级离散化最多可达第五。提出的方法建立了第一种实用方法，该方法提供了针对对流问题的小且可扩展的跨度迭代计数。
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.
