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Volume 8, Issue 4
Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient

Chengming Huang & Shu-Lin Wu

East Asian J. Appl. Math., 8 (2018), pp. 746-763.

Published online: 2018-10

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  • Abstract

The convergence of Parareal-Euler and -LIIIC2 algorithms using the backward Euler method as a $\mathscr{G}$-propagator for the linear problem $U^′ (t)+α(t)A^ηU(t)=f(t)$ with a non-constant coefficient $α$ is studied. We propose to employ the propagator $G$ to a constant model $U^′(t)+βA^ ηU(t)=f(t)$ with a special coefficient β instead of applying both propagators $\mathscr{G}$ and $\mathscr{F}$ to the same target model. We established a simple formula to find an optimal parameter $β_{opt}$, minimising the convergence factor for all mesh ratios. Numerical results confirm the proximity of theoretical optimal $β_{opt}$ to the optimal numerical parameter.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-8-746, author = {Chengming Huang and Shu-Lin Wu}, title = {Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {8}, number = {4}, pages = {746--763}, abstract = {

The convergence of Parareal-Euler and -LIIIC2 algorithms using the backward Euler method as a $\mathscr{G}$-propagator for the linear problem $U^′ (t)+α(t)A^ηU(t)=f(t)$ with a non-constant coefficient $α$ is studied. We propose to employ the propagator $G$ to a constant model $U^′(t)+βA^ ηU(t)=f(t)$ with a special coefficient β instead of applying both propagators $\mathscr{G}$ and $\mathscr{F}$ to the same target model. We established a simple formula to find an optimal parameter $β_{opt}$, minimising the convergence factor for all mesh ratios. Numerical results confirm the proximity of theoretical optimal $β_{opt}$ to the optimal numerical parameter.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.220418.210718}, url = {http://global-sci.org/intro/article_detail/eajam/12817.html} }
TY - JOUR T1 - Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient AU - Chengming Huang & Shu-Lin Wu JO - East Asian Journal on Applied Mathematics VL - 4 SP - 746 EP - 763 PY - 2018 DA - 2018/10 SN - 8 DO - http://doi.org/10.4208/eajam.220418.210718 UR - https://global-sci.org/intro/article_detail/eajam/12817.html KW - Parareal method, time-varying problem, convergence analysis, parameter optimisation. AB -

The convergence of Parareal-Euler and -LIIIC2 algorithms using the backward Euler method as a $\mathscr{G}$-propagator for the linear problem $U^′ (t)+α(t)A^ηU(t)=f(t)$ with a non-constant coefficient $α$ is studied. We propose to employ the propagator $G$ to a constant model $U^′(t)+βA^ ηU(t)=f(t)$ with a special coefficient β instead of applying both propagators $\mathscr{G}$ and $\mathscr{F}$ to the same target model. We established a simple formula to find an optimal parameter $β_{opt}$, minimising the convergence factor for all mesh ratios. Numerical results confirm the proximity of theoretical optimal $β_{opt}$ to the optimal numerical parameter.

Chengming Huang and Shu-Lin Wu. (2018). Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient. East Asian Journal on Applied Mathematics. 8 (4). 746-763. doi:10.4208/eajam.220418.210718
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