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### Numerically modelling stochastic lie transport in fluid dynamics

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StochasticEuler.pdf | Accepted version | 7.04 MB | Adobe PDF | View/Open |

Title: | Numerically modelling stochastic lie transport in fluid dynamics |

Authors: | Cotter, CJ Crisan, D Holm, DD Pan, W Shevchenko, I |

Item Type: | Journal Article |

Abstract: | We present a numerical investigation of stochastic transport in ideal fluids. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. In the current paper, we develop new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation for incompressible 2D Euler fluid flows. The new methodology tested here is found to be suitable for coarse graining in this case. Specifically, we perform uncertainty quantification tests of the velocity decomposition of Cotter et al. (2017), by comparing ensembles of coarse-grid realisations of solutions of the resulting stochastic partial differential equation with the "true solutions" of the deterministic fluid partial differential equation, computed on a refined grid. The time discretization used for approximating the solution of the stochastic partial differential equation is shown to be consistent. We include comprehensive numerical tests that confirm the non-Gaussianity of the stream function, velocity and vorticity fields in the case of incompressible 2D Euler fluid flows. |

Issue Date: | 30-Jan-2019 |

Date of Acceptance: | 20-Nov-2018 |

URI: | http://hdl.handle.net/10044/1/66472 |

DOI: | https://doi.org/10.1137/18M1167929 |

ISSN: | 1064-8275 |

Publisher: | Society for Industrial and Applied Mathematics |

Start Page: | 192 |

End Page: | 232 |

Journal / Book Title: | SIAM Journal on Scientific Computing |

Volume: | 17 |

Issue: | 1 |

Copyright Statement: | © 2019, Society for Industrial and Applied Mathematics |

Sponsor/Funder: | Engineering and Physical Sciences Research Council |

Funder's Grant Number: | EP/N023781/1 |

Keywords: | Science & Technology Physical Sciences Mathematics, Interdisciplinary Applications Physics, Mathematical Mathematics Physics geophysical fluid dynamics stochastic Lie transport uncertainty quantification stochastic partial differential equation 2D Euler equation stochastic parameterization EQUATIONS SPACE TIME physics.flu-dyn physics.flu-dyn 76B99 (primary), 65Z05, 60G99 (secondary) physics.flu-dyn physics.flu-dyn 76B99 (primary), 65Z05, 60G99 (secondary) 0102 Applied Mathematics Applied Mathematics |

Notes: | 41 pages, 26 figures Minor changes -- updated figures to improve readability. Corrected typos. Shifted Remark 7 to just after Assumption A1. Added Remark 8 |

Publication Status: | Published |

Online Publication Date: | 2019-01-30 |

Appears in Collections: | Pure Mathematics Applied Mathematics and Mathematical Physics Faculty of Natural Sciences Mathematics |