Computational Modeling for High-Fidelity Coarsening of Shallow Water Equations Based on Subgrid Data

Abstract

A resolution-independent data-driven stochastic parametrization method for subgrid-scale processes in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale parametrization is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation. The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere. Applying the method at two significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients. Stable and accurate large-scale dynamics can be simulated over long integration times.

Data-driven spectral turbulence modelling for Rayleigh–Bénard convection

“…Frederiksen & Kepert (2006) employed DNS data of the barotropic vorticity equation to model the time series of spherical harmonics as stochastic processes with memory effects, leading to accurate kinetic energy spectra in coarse-grid simulations. Using proper orthogonal decomposition (POD), Ephrati et al (2022b) showed that applying corrections to coarse-grid numerical simulations may lead to significant error reduction. Machine-learning methods have also been successfully employed to find subgrid models (Beck et al 2019), reporting improved results compared to traditional eddy-viscosity models.…”

“…Using proper orthogonal decomposition (POD), Ephrati et al. (2022 b ) showed that applying corrections to coarse-grid numerical simulations may lead to significant error reduction. Machine-learning methods have also been successfully employed to find subgrid models (Beck et al.…”

Data-driven stochastic modeling for coarsened computational geophysical fluid dynamics

“…Data-driven LES methods have successfully been developed in recent years, for example, by using neural networks to compute a variable eddy viscosity [11] to approximate a reference kinetic energy spectrum [101] or to model subgrid-scale forces [170]. Alternatively, approaches based on interpolation of small high-resolution patches of the spatial domain [28,26] and data-driven residual modeling via global basis functions [57] have also shown computational efficiency and accuracy in coarse-grained numerical solutions.…”

“…The material in this chapter was published in the journal Multiscale Modeling and Simulation, see [57].…”

“…In [62] DNS data were employed of the barotropic vorticity equation to model the time series of spherical harmonics as stochastic processes with memory effects, leading to accurate kinetic energy spectra in coarse-grid simulations. Using proper orthogonal decomposition (POD), [57] 5.1. Introduction showed that applying corrections to coarse-grid numerical simulations may lead to significant error reduction.…”