Zeitlin’s model for axisymmetric 3D Euler equations

Abstract Zeitlin’s model is a spatial discretisation for the 2D Euler equations on the flat two-torus or the two-sphere. Contrary to other discretisations, 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 three-sphere. It is the first discretisation of the 3D 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. We also provide numerical experiments to showcase the method.