Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping.

中文翻译：

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优化线性平流的多重网格时间缩减和 Parareal 粗网格算子

在现代高性能计算环境中模拟时间相关偏微分方程 (PDE) 时，时间并行方法，例如多重网格时间缩减 (MGRIT) 和 Parareal，为增加并发性提供了有吸引力的选择。虽然这些技术对于抛物线方程非常成功，但经常观察到，当应用于对流主导问题或在粗网格上使用标准重新离散化方法的纯双曲线 PDE 时，它们的性能会受到严重影响。在本文中，我们将 MGRIT 或 Parareal 应用于常数系数线性平流方程，利用现有的收敛理论来深入了解这些求解器在解决此问题时通常不可缩放甚至发散的行为。为了克服这些缺点，我们用改进的粗网格算子代替了粗网格上的重新离​​散化，这些算子是通过应用优化技术来计算的，以近似最小化收敛理论中的误差估计。我们的主要发现之一是，为了获得抛物线问题的快速收敛，粗网格算子应该通过跟踪特征曲线来考虑双曲线问题的行为。我们的方法使用显式或隐式 Runge-Kutta 方法结合迎风有限差分空间离散化来测试各种阶的方案。在所有情况下，我们只需几次迭代即可获得可扩展的收敛性，并行测试也显示出相对于顺序时间步长的显着加速。