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 multigrid 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 are compared for varying orders of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences in execution times. For the linear problem, we observe that the two methods show similar scaling behavior with PFASST being more favorable for high order methods or when few parallel resources are available. For the non-linear problem, PFASST is more flexible in terms of solution strategy, while space-time multigrid requires a full non-linear solve.

我们考虑应用于（反应）扩散问题的两种时间并行方法，可能是非线性的。特别是，我们考虑了 PFASST（时空并行完全逼近方案）和时空多重网格策略。对于这两种方法，我们从连续时间相关问题的积分公式开始。然后，采用 PFASST 的搭配形式和时空多重网格的时间不连续 Galerkin 离散化，在时间节点上产生相同的离散解。针对时间离散化的不同阶数，比较了两种多级策略的强和弱缩放。此外，我们研究了非线性问题的各自收敛行为，并突出了执行时间的定量差异。对于线性问题，我们观察到这两种方法显示出类似的缩放行为，PFASST 更适合高阶方法或当可用的并行资源很少时。对于非线性问题，PFASST在求解策略上更加灵活，而时空多重网格则需要完全非线性求解。