scholarly journals Optimized Schwarz Waveform Relaxation Algorithms with Nonconforming Time Discretization for Coupling Convection-diffusion Problems with Discontinuous Coefficients

2007 ◽  
pp. 267-274 ◽  
Author(s):  
Eric Blayo ◽  
Laurence Halpern ◽  
Caroline Japhet
Keyword(s):  
Time Discretization ◽  
Discontinuous Coefficients ◽  
Waveform Relaxation ◽  
Convection Diffusion ◽  
Schwarz Waveform Relaxation ◽  
Relaxation Algorithms ◽  
Diffusion Problems

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

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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