The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and analyze two parareal algorithms based on the Dirichlet-Neumann and the Neumann-Neumann waveform relaxation method for the heat equation by choosing Dirichlet-Neumann/Neumann-Neumann waveform relaxation as two new kinds of fine propagators instead of the classical fine propagator. Both new proposed algorithms could be viewed as a space-time parallel algorithm, which increases the parallelism both in space and in time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments finally.

Dirichlet-Neumann和Neumann-Neumann波形松弛方法是用于解决演化问题的非重叠空间域分解方法，而超现实算法是时间并行的。基于这些时空并行策略的组合，通过选择Dirichlet-Neumann / Neumann-Neumann波形弛豫为两个，提出并分析了两种基于Dirichlet-Neumann和Neumann-Neumann波形弛豫法的热方程的超现实算法。新型的精细传播器，而不是经典的精细传播器。两种新提出的算法都可以看作是时空并行算法，它在空间和时间上都增加了并行性。我们为热方程推导两种算法在一个空间维度上的收敛结果。