NUMERICALLY MODELING STOCHASTIC LIE TRANSPORT IN FLUID DYNAMICS

We present a numerical investigation of stochastic transport in ideal fluids. According to Holm [Proc. A, 471 (2015)] and Cotter, Gottwald, and Holm [Proc. A, 473 (2017)], the principles of transformation theory and multitime homogenization, 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 a new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretization for incompressible two-dimensional (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, Gottwald, and Holm, by comparing ensembles of coarse grid realizations 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 con firm the non-Gaussianity of the streamfunction, velocity, and vorticity fields in the case of incompressible 2D Euler fluid flows.