Convergence Results for Non-Overlap Schwarz Waveform Relaxation Algorithm with Changing Transmission Conditions

Heat equations with distributed delay are a class of mathematic models that has wide applications in many fields. Numerical computation plays an important role in the investigation of these equations, because the analytic solutions of partial differential equations with time delay are usually unavailable. On the other hand, duo to the delay property, numerical computation of these equations is time-consuming. To reduce the computation time, we analyze in this paper the Schwarz waveform relaxation algorithm with Robin transmission conditions. The Robin transmission conditions contain a free parameter, which has a significant effect on the convergence rate of the Schwarz waveform relaxation algorithm. Determining the Robin parameter is therefore one of the top-priority matters for the study of the Schwarz waveform relaxation algorithm. We provide new formula to fix the Robin parameter and we show numerically that the new Robin parameter is more efficient than the one proposed previously in the literature.

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Nonoverlapping Schwarz Waveform Relaxation Algorithm for a Class of Time-Fractional Heat Equations

Fundamenta Informaticae ◽

10.3233/fi-2017-1489 ◽

2017 ◽

Vol 151 (1-4) ◽

pp. 231-240

Author(s):

Shu-Lin Wu ◽

Guo-Cheng Wu

Keyword(s):

Waveform Relaxation ◽

Heat Equations ◽

Relaxation Algorithm ◽

Schwarz Waveform Relaxation

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Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2018-0188 ◽

2020 ◽

Vol 20 (3) ◽

pp. 397-417

Author(s):

Mohammad Al-Khaleel ◽

Shu-Lin Wu

Keyword(s):

Transmission Conditions ◽

Parabolic Problems ◽

Waveform Relaxation ◽

Discrete Level ◽

Convergence Factor ◽

Schwarz Waveform Relaxation ◽

Relaxation Algorithms ◽

Time Dependent Problems ◽

New Algorithms ◽

Level Analysis

AbstractThe 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.

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