We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.

我们讨论如何设计“空间并行和时间同时”牛顿多重网格方法，该方法通过减少等待时间成本来改善空间并行性的缩放行为。这个想法是立即解决许多时间步长，因此解决的问题是数量较少但规模较大的系统。这些大型系统被重新排序，并被解释为仅空间问题，从而导致在空间上半粗化并在时间方向上进行线平滑的多重网格算法。通过将其作为前置条件嵌入Krylov子空间方法中，可以进一步改进该平滑器。作为一种典型的应用，我们关注于具有多达数千个时间步长的标量偏微分方程（PDE），这些时间步长在时间，空间，空间上通过有限差分，空间有限元方法离散化。对于线性PDE，所得方法与多重网格波形弛豫及其理论框架密切相关。在我们的抛物线测试问题中，这种多网格方法的数值行为在空间和时间网格大小以及同时处理的时间步数方面都很稳健。此外，我们说明了如何为需要在每个外部非线性步骤中描述线性化问题的解的非线性非平稳问题导出对应的时间同时定点解和牛顿型解。作为主要结果，我们能够生成大得多的问题，需要大量核处理，因此，稳健缩放的多网格求解器与更大程度的并行度相结合，可以更快地解决非平稳问题。