Numerically Modelling Stochastic Lie Transport in Fluid Dynamics

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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: 1-Jan-2019
Date of Acceptance: 20-Nov-2018
ISSN: 1064-8275
Publisher: Society for Industrial and Applied Mathematics
Journal / Book Title: SIAM Journal on Scientific Computing
Copyright Statement: This paper is embargoed until publication.
Sponsor/Funder: Engineering and Physical Sciences Research Council
Funder's Grant Number: EP/N023781/1
Keywords: physics.flu-dyn
76B99 (primary), 65Z05, 60G99 (secondary)
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
0802 Computation Theory And Mathematics
Numerical & Computational 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
Embargo Date: publication subject to indefinite embargo
Appears in Collections:Pure Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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