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Volume 15, Issue 4-5
A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations

Max Gunzburger, Nan Jiang & Michael Schneier

Int. J. Numer. Anal. Mod., 15 (2018), pp. 608-627.

Published online: 2018-04

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  • Abstract

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) approach was incorporated into a first-order accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a second-order accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM approach. Numerical experiments illustrate the accuracy and efficiency of the new approach.

  • Keywords

Navier-Stokes equations, ensemble computation, proper orthogonal decomposition, finite element methods.

  • AMS Subject Headings

35Q30, 65N30, 65M99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mgunzburger@fsu.edu (Max Gunzburger)

jiangn@mst.edu (Nan Jiang)

mschneier89@gmail.com (Michael Schneier)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-608, author = {Gunzburger , MaxJiang , Nan and Schneier , Michael}, title = {A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {4-5}, pages = {608--627}, abstract = {

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) approach was incorporated into a first-order accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a second-order accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM approach. Numerical experiments illustrate the accuracy and efficiency of the new approach.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12534.html} }
TY - JOUR T1 - A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations AU - Gunzburger , Max AU - Jiang , Nan AU - Schneier , Michael JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 608 EP - 627 PY - 2018 DA - 2018/04 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12534.html KW - Navier-Stokes equations, ensemble computation, proper orthogonal decomposition, finite element methods. AB -

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) approach was incorporated into a first-order accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a second-order accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM approach. Numerical experiments illustrate the accuracy and efficiency of the new approach.

Max Gunzburger, Nan Jiang & Michael Schneier. (2020). A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations. International Journal of Numerical Analysis and Modeling. 15 (4-5). 608-627. doi:
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