We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time-dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.

翻译：我们认为,对一个(反应)扩散问题适用两种平行时间办法,可能是非线性办法,特别是,我们考虑了PFASST(空间和时间全面接近计划)和时空多层次战略。对于这两种办法,我们从持续的时间依赖问题的整体形成开始。然后,对PFASST采用了同位形式,对时空多电网采用了不连续的加勒金分解形式,结果在时节点出现同样的离散解决办法。对两个多层次战略的强弱规模进行了对比,以不同时间分解顺序进行比较。此外,我们调查非线性问题的趋同行为,并突出数量差异。