Space-time methods are able to solve time-dependent problems faster by exploiting the full power of high-performance computer. In this article, a multilevel space-time multiplicative Schwarz method is presented for solving parabolic equations in parallel on both spatial and temporal directions. In the implementation, the proposed Schwarz method is treated as preconditioner for GMRES, that is, a coupled system arising from the discretization of the parabolic equation is solved by using a multiplicative Schwarz preconditioned GMRES algorithm. We develop an optimal convergence theory to show that the convergence rate is bounded and independent of the spatial mesh sizes, the time step size, the number of subdomains, the number of levels, and the window size. Some numerical results obtained on a parallel computer with thousands of processors are presented to confirm the theory in terms of optimality and scalability. Moreover, numerical comparisons with traditional time-stepping algorithms show that the proposed space-time method earns lots of benefits when the number of processor cores is large.

中文翻译：

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抛物线方程的多级时空乘法 Schwarz 预处理器

时空方法能够通过充分利用高性能计算机的全部功能更快地解决与时间相关的问题。在本文中，提出了一种多级时空乘法 Schwarz 方法，用于在空间和时间方向上并行求解抛物线方程。在实现中，将所提出的Schwarz方法作为GMRES的预处理器，即使用乘法Schwarz预处理GMRES算法求解由抛物线方程离散化产生的耦合系统。我们开发了一个最优收敛理论来表明收敛速度是有界的，并且与空间网格大小、时间步长、子域数量、级别数量和窗口大小无关。给出了在具有数千个处理器的并行计算机上获得的一些数值结果，以在最优性和可扩展性方面证实该理论。此外，与传统时间步进算法的数值比较表明，当处理器内核数量很大时，所提出的时空方法可以获得很多好处。