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Volume 25, Issue 2
Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations

Shu-Lin Wu & Chengming Huang

DOI: 10.4208/cicp.OA-2017-0177

Commun. Comput. Phys., 25 (2019), pp. 390-415.

Published online: 2018-10

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  • Abstract

The equioscillation principle is an important rule to fix the parameter for the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions. For parabolic PDEs with integer order temporal derivative, such a principle yields optimal Robin parameter, while in our previous study we found numerically that it is not always the case for time fractional PDEs: the Robin parameter determined by the equioscillation principle is sometimes far away from optimal. In this paper, by using the time fractional Cable equations as the model, we show that our previous finding does not happen occasionally but an inherent property of the SWR algorithm. Our analysis also reveals an essential difference between the asymptotic convergence rates in the overlapping and non-overlapping cases. Numerical results are provided to validate our theoretical analysis.

  • Keywords

Schwarz waveform relaxation, fractional Cable equation, parameter optimization, asymptotic analysis.

  • AMS Subject Headings

65M55, 65M12, 65M15, 65Y05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-390, author = {Shu-Lin Wu and Chengming Huang}, title = {Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {2}, pages = {390--415}, abstract = {

The equioscillation principle is an important rule to fix the parameter for the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions. For parabolic PDEs with integer order temporal derivative, such a principle yields optimal Robin parameter, while in our previous study we found numerically that it is not always the case for time fractional PDEs: the Robin parameter determined by the equioscillation principle is sometimes far away from optimal. In this paper, by using the time fractional Cable equations as the model, we show that our previous finding does not happen occasionally but an inherent property of the SWR algorithm. Our analysis also reveals an essential difference between the asymptotic convergence rates in the overlapping and non-overlapping cases. Numerical results are provided to validate our theoretical analysis.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0177}, url = {http://global-sci.org/intro/article_detail/cicp/12756.html} }
TY - JOUR T1 - Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations AU - Shu-Lin Wu & Chengming Huang JO - Communications in Computational Physics VL - 2 SP - 390 EP - 415 PY - 2018 DA - 2018/10 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0177 UR - https://global-sci.org/intro/article_detail/cicp/12756.html KW - Schwarz waveform relaxation, fractional Cable equation, parameter optimization, asymptotic analysis. AB -

The equioscillation principle is an important rule to fix the parameter for the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions. For parabolic PDEs with integer order temporal derivative, such a principle yields optimal Robin parameter, while in our previous study we found numerically that it is not always the case for time fractional PDEs: the Robin parameter determined by the equioscillation principle is sometimes far away from optimal. In this paper, by using the time fractional Cable equations as the model, we show that our previous finding does not happen occasionally but an inherent property of the SWR algorithm. Our analysis also reveals an essential difference between the asymptotic convergence rates in the overlapping and non-overlapping cases. Numerical results are provided to validate our theoretical analysis.

Shu-Lin Wu and Chengming Huang. (2018). Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations. Communications in Computational Physics. 25 (2). 390-415. doi:10.4208/cicp.OA-2017-0177
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