Geometric low-rank approximation of the Zeitlin model of incompressible fluids on the sphere

TL;DR

This paper introduces a low-rank approximation method for Zeitlin's geometric model of 2D incompressible fluids on the sphere, preserving key structures and reducing computational complexity.

Contribution

It proposes a novel low-rank factorization approach that maintains the isospectral Lie--Poisson structure of Zeitlin's model, improving efficiency.

Findings

The approximate flow remains isospectral and Lie--Poisson.

Error depends only on initial vorticity matrix approximation.

Computational complexity scales quadratically with matrix order.

Abstract

We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie--Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin's model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie--Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.