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:在平行时间群体中,一个长期的问题是双曲部分差异方程式(PDEs)的标准迭代平行时间方法与超曲偏偏偏偏偏差方程式(PDEs)的标准迭代平行时间方法的趋同性差,以及更广义的对调占主导地位的PDEs。在这里,对迭代平行减少多格在时间(MGRIIT)的多格减少(MGRIIT)模式的双级平行时间方法(LFA)的趋同性,产生了一种局部的Fourier分析(LFA)趋同性理论(LFA)的趋同性差。这一封闭式理论使人们可以重新洞见地发现,当使用重分解偏差的微网格方程式对准(PDE)标准方法重新分解时,MGRIT的趋同性平行方法的趋同性差性差,至少部分是由于某些被称为特征组成部分的平流四面法模式的粗略的统合网格修正(LFAGRIT)的趋同性模式不力。 我们采用这种趋同性理论理论理论来证明,在不断改进的CRIGRIGRICT中可以使这种分解的极地分析中转的理论推理法的极法的理论的分解法的理论,这种分解法的分法。