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# Advances in Spectral Methods for UQ in Incompressible Navier-Stokes Equations

- Handle URI:
- http://hdl.handle.net/10754/624018
- Title:
- Advances in Spectral Methods for UQ in Incompressible Navier-Stokes Equations
- Authors:
- Abstract:
- In this talk, I will present two recent contributions to the development of efficient methodologies for uncertainty propagation in the incompressible Navier-Stokes equations. The first one concerns the reduced basis approximation of stochastic steady solutions, using Proper Generalized Decompositions (PGD). An Arnoldi problem is projected to obtain a low dimensional Galerkin problem. The construction then amounts to the resolution of a sequence of uncoupled deterministic Navier-Stokes like problem and simple quadratic stochastic problems, followed by the resolution of a low-dimensional coupled quadratic stochastic problem, with a resulting complexity which has to be contrasted with the dimension of the whole Galerkin problem for classical spectral approaches. An efficient algorithm for the approximation of the stochastic pressure field is also proposed. Computations are presented for uncertain viscosity and forcing term to demonstrate the effectiveness of the reduced method. The second contribution concerns the computation of stochastic periodic solutions to the Navier-Stokes equations. The objective is to circumvent the well-known limitation of spectral methods for long-time integration. We propose to directly determine the stochastic limit-cycles through the definition of its stochastic period and an initial condition over the cycle. A modified Newton method is constructed to compute iteratively both the period and initial conditions. Owing to the periodic character of the solution, and by introducing an appropriate time-scaling, the solution can be approximated using low-degree polynomial expansions with large computational saving as a result. The methodology is illustrated for the von-Karman flow around a cylinder with stochastic inflow conditions.
- KAUST Department:
- Conference/Event name:
- Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)
- Issue Date:
- 6-Jan-2014
- Type:
- Presentation

- Appears in Collections:
- Presentations; Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Le Maitre, Olivier | en |

dc.date.accessioned | 2017-06-01T10:20:43Z | - |

dc.date.available | 2017-06-01T10:20:43Z | - |

dc.date.issued | 2014-01-06 | - |

dc.identifier.uri | http://hdl.handle.net/10754/624018 | - |

dc.description.abstract | In this talk, I will present two recent contributions to the development of efficient methodologies for uncertainty propagation in the incompressible Navier-Stokes equations. The first one concerns the reduced basis approximation of stochastic steady solutions, using Proper Generalized Decompositions (PGD). An Arnoldi problem is projected to obtain a low dimensional Galerkin problem. The construction then amounts to the resolution of a sequence of uncoupled deterministic Navier-Stokes like problem and simple quadratic stochastic problems, followed by the resolution of a low-dimensional coupled quadratic stochastic problem, with a resulting complexity which has to be contrasted with the dimension of the whole Galerkin problem for classical spectral approaches. An efficient algorithm for the approximation of the stochastic pressure field is also proposed. Computations are presented for uncertain viscosity and forcing term to demonstrate the effectiveness of the reduced method. The second contribution concerns the computation of stochastic periodic solutions to the Navier-Stokes equations. The objective is to circumvent the well-known limitation of spectral methods for long-time integration. We propose to directly determine the stochastic limit-cycles through the definition of its stochastic period and an initial condition over the cycle. A modified Newton method is constructed to compute iteratively both the period and initial conditions. Owing to the periodic character of the solution, and by introducing an appropriate time-scaling, the solution can be approximated using low-degree polynomial expansions with large computational saving as a result. The methodology is illustrated for the von-Karman flow around a cylinder with stochastic inflow conditions. | en |

dc.title | Advances in Spectral Methods for UQ in Incompressible Navier-Stokes Equations | en |

dc.type | Presentation | en |

dc.contributor.department | Computer, Electrical and Mathematical Sciences & Engineering (CEMSE) | en |

dc.conference.date | January 6-10, 2014 | en |

dc.conference.name | Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014) | en |

dc.conference.location | KAUST | en |

kaust.author | Le Maitre, Olivier | en |

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