Convergence analysis for discrete Schwarz waveform relaxation algorithm of Robin type

2013 ◽  
Vol 43 (3) ◽  
pp. 211-234 ◽  
Author(s):  
ShuLin WU
Keyword(s):  
Convergence Analysis ◽  
Waveform Relaxation ◽  
Relaxation Algorithm ◽  
Schwarz Waveform Relaxation

scholarly journals Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Ke Li ◽  
Dali Guo ◽  
Yunxiang Zhao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Good Choice ◽  
Nonlocal Diffusion ◽  
Waveform Relaxation ◽  
Partial Differential ◽  
Schwarz Waveform Relaxation ◽  
Singular Kernels ◽  
The One

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

scholarly journals Schwarz waveform relaxation algorithm for heat equations with distributed delay

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 659-667
Author(s):  
Shu-Lin Wu
Keyword(s):  
Numerical Computation ◽  
Distributed Delay ◽  
Computation Time ◽  
Analytic Solutions ◽  
Transmission Conditions ◽  
Waveform Relaxation ◽  
Heat Equations ◽  
Relaxation Algorithm ◽  
Schwarz Waveform Relaxation ◽  
The One

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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