We investigate three directions to further improve the highly efficient Space-Time Multigrid algorithm with block-Jacobi smoother introduced in [GanNeu16]. First, we derive an analytical expression for the optimal smoothing parameter in the case of a full space-time coarsening strategy; second, we propose a new and efficient direct coarsening strategy which simplifies the code by preventing changes of coarsening regimes; and third, we also optimize the entire two cycle to investigate if further efficiency gains are possible. Especially, we show that our new coarsening strategy leads to a significant efficiency gain when the ratio $\tau/h^2$ is small, where $\tau$ and $h$ represent the time and space steps. Our analysis is performed for the heat equation in one spatial dimension, using centered finite differences in space and Backward Euler in time, but could be generalized to other situations. We also present numerical experiments that confirm our theoretical findings.
翻译:我们调查了三个方向,以进一步改进在[GanNeu16] 中引入的高效空间-时多格丽德算法,该算法以组合-Jacobi平滑法来进一步改进。首先,我们为全时全时粗化战略的最佳平滑参数进行了分析表达;第二,我们提出了新的高效直接粗化战略,通过防止粗化制度的变化来简化代码;第三,我们还优化了整个两个周期,以在可能进一步提高效率的情况下进行调查。特别是,我们表明,我们新的粗化战略在美元/小时/2美元比率很小,即美元和美元代表时间和空间步骤时,将带来显著的效率收益。我们的分析是在一个空间层面进行,利用空间和后向的中点差异,但可以在时间上推广到其他情况下。我们还进行了数字实验,以证实我们的理论结论。</s>