Zeitlin's model for axisymmetric 3-D Euler equations

TL;DR

This paper extends Zeitlin's geometric discretization method from 2-D to axisymmetric 3-D Euler equations on the 3-sphere, preserving the geometric structure and enabling analysis of curvature and stability.

Contribution

It introduces the first geometric structure-preserving discretization for axisymmetric 3-D Euler equations on the 3-sphere.

Findings

Preserves the geometric structure of the Euler equations

Allows analysis of Riemannian curvature and Jacobi equations

First discretization of 3-D Euler equations with full geometric preservation

Abstract

Zeitlin's model is a spatial discretization for the 2-D Euler equations on the flat 2-torus or the 2-sphere. Contrary to other discretizations, it preserves the underlying geometric structure, namely that the Euler equations describe Riemannian geodesics on a Lie group. Here we show how to extend Zeitlin's approach to the axisymmetric Euler equations on the 3-sphere. It is the first discretization of the 3-D Euler equations that fully preserves the geometric structure, albeit restricted to axisymmetric solutions. Thus, this finite-dimensional model admits Riemannian curvature and Jacobi equations, which are discussed.