Numerically Modelling Stochastic Lie Transport in Fluid Dynamics

TL;DR

This paper develops and tests a numerical method for decomposing fluid transport velocities into deterministic and stochastic components, enabling uncertainty quantification in 2D Euler fluid flows.

Contribution

It introduces a new methodology for velocity decomposition and stochastic PDE integration, suitable for coarse graining and uncertainty quantification in fluid dynamics.

Findings

The methodology accurately captures stochastic effects in fluid transport.

Ensemble simulations match refined grid solutions, confirming validity.

Fields exhibit non-Gaussian behavior in 2D Euler flows.

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.