1. Numerically Modeling Stochastic Lie Transport in Fluid Dynamics
- Author
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Wei Pan, Darryl D. Holm, Colin J. Cotter, Dan Crisan, Igor Shevchenko, and Engineering and Physical Sciences Research Council
- Subjects
Mathematics, Interdisciplinary Applications ,uncertainty quantification ,FOS: Physical sciences ,General Physics and Astronomy ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,76B99 (primary), 65Z05, 60G99 (secondary) ,Physics::Fluid Dynamics ,Geophysical fluid dynamics ,0102 Applied Mathematics ,geophysical fluid dynamics ,Fluid dynamics ,SPACE ,Applied mathematics ,0101 mathematics ,Uncertainty quantification ,EQUATIONS ,stochastic parameterization ,Physics ,Science & Technology ,Ideal (set theory) ,2D Euler equation ,Applied Mathematics ,Ecological Modeling ,Fluid Dynamics (physics.flu-dyn) ,Physics - Fluid Dynamics ,General Chemistry ,stochastic partial differential equation ,TIME ,Computer Science Applications ,Physics, Mathematical ,010101 applied mathematics ,Stochastic partial differential equation ,physics.flu-dyn ,Modeling and Simulation ,Physical Sciences ,stochastic Lie transport ,Mathematics - 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., 41 pages, 26 figures Minor changes -- updated figures to improve readability. Corrected typos. Shifted Remark 7 to just after Assumption A1. Added Remark 8
- Published
- 2019