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.

scholarly journals OPTIMIZED AND QUASI-OPTIMAL SCHWARZ WAVEFORM RELAXATION FOR THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2010 ◽  
Vol 20 (12) ◽  
pp. 2167-2199 ◽  
Author(s):  
LAURENCE HALPERN ◽  
JÉRÉMIE SZEFTEL
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Optimal Algorithm ◽  
Waveform Relaxation ◽  
One Dimensional ◽  
Schwarz Waveform Relaxation ◽  
Discrete Algorithms ◽  
Relaxation Algorithms ◽  
The One

Schwarz waveform relaxation algorithms are designed for the linear Schrödinger equation with potential. Two classes of algorithms are introduced: the quasi-optimal algorithm, based on the transparent continuous or discrete boundary condition, and the optimized complex Robin algorithm. We analyze their properties in one dimension. First, well-posedness and convergence are studied, in the overlapping and the non-overlapping case, for constant or non-constant potentials. Then discrete algorithms are established, for which convergence is proved through discrete energies or Fourier transforms, as in the continuous case. Numerical results illustrate the efficiency of the methods, for various types of potentials and any number of subdomains.

scholarly journals Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations

2010 ◽  
Vol 5 (3) ◽  
pp. 487-505 ◽  
Author(s):  
Filipa Caetano ◽  
◽  
Martin J. Gander ◽  
Laurence Halpern ◽  
Jérémie Szeftel ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Waveform Relaxation ◽  
Schwarz Waveform Relaxation ◽  
Relaxation Algorithms
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