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Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem
Computational Methods in Applied Mathematics ( IF 1.0 ) Pub Date : 2020-07-01 , DOI: 10.1515/cmam-2018-0188
Mohammad Al-Khaleel 1 , Shu-Lin Wu 2
Affiliation  

Abstract The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level. In this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones. We shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods. We first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence. Numerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy.

中文翻译:

准重叠半离散 Schwarz 波形松弛算法:双曲线问题

摘要 Schwarz 波形弛豫 (SWR) 算法具有许多有利的特性,并被广泛研究和研究以解决主要在连续级别上的时间相关问题。在本文中,我们考虑了半离散级分析,并研究了所谓的半离散 SWR 算法与离散传输条件相结合而不是连续传输条件的收敛行为。我们将在这里针对双曲线问题,而不是大多数研究人员在研究 SWR 方法的性质时通常考虑的抛物线问题。我们首先展示了具有不同分区选择的经典重叠半离散 SWR 算法,并表明它们收敛速度非常慢。然后我们引入最优的、优化的、和准优化重叠半离散 SWR 算法使用新的传输条件,也有不同的分区选择。我们表明,通过在半离散级别使用与优化 SWR 算法相关的离散传输条件,新算法会导致更好的收敛。在执行的半离散级别分析中,我们还证明了这样一个事实,即随着重叠大小与空间离散化大小之间的比率变大,收敛因子变小,从而导致更好的收敛。给出了数值结果和实验,以确认所提出算法的理论方面,并提供其有用性和准确性的证据。我们表明,通过在半离散级别使用与优化 SWR 算法相关的离散传输条件,新算法会导致更好的收敛。在执行的半离散级别分析中,我们还证明了这样一个事实,即随着重叠大小与空间离散化大小之间的比率变大,收敛因子变小,从而导致更好的收敛。给出了数值结果和实验,以确认所提出算法的理论方面,并提供其有用性和准确性的证据。我们表明,通过在半离散级别使用与优化 SWR 算法相关的离散传输条件,新算法可以实现更好的收敛。在执行的半离散级别分析中,我们还证明了这样一个事实,即随着重叠大小与空间离散化大小之间的比率变大,收敛因子变小,从而导致更好的收敛。给出了数值结果和实验,以确认所提出算法的理论方面,并提供其有用性和准确性的证据。我们还证明了这样一个事实:随着重叠大小和空间离散化大小之间的比率变大,收敛因子变小,从而导致更好的收敛。给出了数值结果和实验,以确认所提出算法的理论方面,并提供其有用性和准确性的证据。我们还证明了这样一个事实:随着重叠大小和空间离散化大小之间的比率变大,收敛因子变小,从而导致更好的收敛。给出了数值结果和实验,以确认所提出算法的理论方面,并提供其有用性和准确性的证据。
更新日期:2020-07-01
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