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：我们研究了三个方向以进一步改进[GanNeu16]中提出的基于块-Jacobi光滑器的高效时空多重网格算法。首先，在全时空粗化策略下，我们推导了最优光滑参数的解析表达式；其次，提出了一种新的高效直接粗化策略，该策略通过避免粗化模式切换来简化代码实现；最后，我们对整个两重网格循环进行优化，以探究进一步提升效率的可能性。特别地，我们证明了当参数$\tau/h^2$（其中$\tau$和$h$分别代表时间步长和空间步长）较小时，所提出的粗化策略能显著提升效率。本分析基于一维空间热传导方程，采用空间中心差分与时间向后欧拉格式，但可推广至其他情形。我们还通过数值实验验证了理论结果的正确性。