With the advent of massively parallel computers with thousands of processors, a large amount of work has been done during the last decades in order to enable a more effective use of a higher number of processors, by superposing parallelism in time-domain, even though it is known that time-integration is inherently sequential, to parallelism in the space-domain [8]. Consequently, many families of predictor-corrector methods have been proposed, allowing computing on several time-steps concurrently [5], [6]. The aim of our present work is to develop a new parallel-in-time algorithm for solving evolution problems, based on particularities of a rescaling method that has been developed for solving different types of partial and ordinary differential equations whose solutions have a finite existence time [9]. Such method leads to a sliced-time computing technique used to solve independently rescaled models of the differential equation. The determining factor for convergence of the iterative process are the predicted values at the start of each time slice. These are obtained using “ratio-based” formulae. In this paper we extend successfully this method to reaction diffusion problems of the form ut=Δ um+aup, with their solutions having a global existence time when p≤ m≤ 1. The resulting algorithm RaPTI provides perfect parallelism, with convergence being reached after few iterations.
展开▼
机译:随着具有数千个处理器的大型平行计算机的出现,在过去的几十年中已经完成了大量的工作,以便通过在时域中的叠加并行性,使得更有效地利用更多数量的处理器,即使它众所周知,时间集成是固有的顺序,在空间域中的并行性[8]。因此,已经提出了许多预测器校正器方法的家庭,允许同时计算几步[5],[6]。我们目前的作品的目的是开发一种新的并行时间算法,用于解决演化问题,基于已经开发的用于解决解决方案具有有限存在时间的不同类型和常微分方程的重新分配方法的特殊性的特殊性[9]。这种方法导致用于求解微分方程的独立重复模型的切片时间计算技术。迭代过程的收敛因素是每次切片开始时的预测值。这些是使用“基于比率”的公式获得的。在本文中,我们成功地扩展了这种方法来实现了FORM UT =ΔMUM+ AUP的反应扩散问题,它们的解决方案在P≤M≤1时具有全局存在时间。结果算法RAPTI提供完美的并行性,以后达到收敛很少的迭代。
展开▼