Sparse-stochastic model reduction for 2D Euler equations

TL;DR

This paper introduces a novel stochastic model reduction method for 2D Euler equations, combining recent techniques to efficiently simulate large-scale fluid dynamics while accounting for small-scale effects.

Contribution

It develops a new approach that integrates stochastic model reduction with conservative semi-discretization based on the Zeitlin model, enabling efficient large-scale simulations of 2D Euler flows.

Findings

Small scales have negligible influence on large scales after turbulence develops.

The method captures energy transfer among modes despite reduced complexity.

Numerical results validate the effectiveness of the stochastic reduction approach.

Abstract

The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several techniques to obtain fast and accurate simulations have been developed during the last decades. In this paper, we present a novel approach which combines recent developments in the stochastic model reduction and conservative semi-discretization of the Euler equations. In particular, starting from the Zeitlin model on the 2-sphere, we derive reduced dynamics for large scales and we close the equations either deterministically or with a suitable stochastic term. Numerical experiments show that, after an initial turbulent regime, the influence of small scales to large scales is negligible, even though a non-zero transfer of energy among different modes is present.