Semi-discrete Schwarz waveform relaxation algorithms for reaction diffusion equations

2014 ◽  
Vol 54 (3) ◽  
pp. 831-866 ◽  
Author(s):  
Shu-Lin Wu ◽  
Mohammad D. Al-Khaleel
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Waveform Relaxation ◽  
Schwarz Waveform Relaxation ◽  
Relaxation Algorithms

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

Quasi-Optimized Overlapping Schwarz Waveform Relaxation Algorithm for PDEs with Time-Delay

2013 ◽  
Vol 14 (3) ◽  
pp. 780-800 ◽  
Author(s):  
Shu-Lin Wu ◽  
Ting-Zhu Huang
Keyword(s):  
Time Delay ◽  
Reaction Diffusion ◽  
Transmission Condition ◽  
Reaction Diffusion Equations ◽  
Complete Analysis ◽  
Waveform Relaxation ◽  
Schwarz Waveform Relaxation ◽  
Regular Time ◽  
The One ◽  
Time Dependent Problems

AbstractSchwarz waveform relaxation (SWR) algorithm has been investigated deeply and widely for regular time dependent problems. But for time delay problems, complete analysis of the algorithm is rare. In this paper, by using the reaction diffusion equations with a constant discrete delay as the underlying model problem, we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition. The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem, which is more complex than the one arising for regular problems without delay. We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter, which is shown efficient to accelerate the convergence of the SWR algorithm. Numerical results are provided to validate the theoretical conclusions.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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