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Volume 20, Issue 3
A Doubly Adaptive Penalty Method for the Navier Stokes Equations

Kiera Kean, Xihui Xie & Shuxian Xu

DOI: 10.4208/ijnam2023-1017

Int. J. Numer. Anal. Mod., 20 (2023), pp. 407-436.

Published online: 2023-03

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  • Abstract

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $ϵ,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $ϵ(t_{n+1}) − ϵ(t_n).$ The analysis and tests show that adapting $ϵ(t_{n+1})$ in response to $∇·u(t_n)$ removes the problem of picking $ϵ$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$ϵ$ method by also adapting the time-step. The penalty parameter ϵ and time-step are adapted independently. We further compare first, second and variable order time-step algorithms. Accurate recovery of pressure remains an open problem.

  • Keywords

Navier-Stokes equations, penalty, adaptive.

  • AMS Subject Headings

65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-407, author = {Kean , KieraXie , Xihui and Xu , Shuxian}, title = {A Doubly Adaptive Penalty Method for the Navier Stokes Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {3}, pages = {407--436}, abstract = {

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $ϵ,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $ϵ(t_{n+1}) − ϵ(t_n).$ The analysis and tests show that adapting $ϵ(t_{n+1})$ in response to $∇·u(t_n)$ removes the problem of picking $ϵ$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$ϵ$ method by also adapting the time-step. The penalty parameter ϵ and time-step are adapted independently. We further compare first, second and variable order time-step algorithms. Accurate recovery of pressure remains an open problem.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1017}, url = {http://global-sci.org/intro/article_detail/ijnam/21540.html} }
TY - JOUR T1 - A Doubly Adaptive Penalty Method for the Navier Stokes Equations AU - Kean , Kiera AU - Xie , Xihui AU - Xu , Shuxian JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 407 EP - 436 PY - 2023 DA - 2023/03 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1017 UR - https://global-sci.org/intro/article_detail/ijnam/21540.html KW - Navier-Stokes equations, penalty, adaptive. AB -

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $ϵ,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $ϵ(t_{n+1}) − ϵ(t_n).$ The analysis and tests show that adapting $ϵ(t_{n+1})$ in response to $∇·u(t_n)$ removes the problem of picking $ϵ$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$ϵ$ method by also adapting the time-step. The penalty parameter ϵ and time-step are adapted independently. We further compare first, second and variable order time-step algorithms. Accurate recovery of pressure remains an open problem.

Kiera Kean, Xihui Xie & Shuxian Xu. (2023). A Doubly Adaptive Penalty Method for the Navier Stokes Equations. International Journal of Numerical Analysis and Modeling. 20 (3). 407-436. doi:10.4208/ijnam2023-1017
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