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.

翻译：我们考虑描述球面上二维不可压缩理想流体流动的欧拉方程涡量形式。Zeitlin模型提供了一种保持底层几何结构的涡量形式有限维逼近：它由斜厄米矩阵李代数上的等谱李-泊松流构成。我们提出基于涡量矩阵的时变低秩分解来逼近Zeitlin模型，并按照欧拉方程演化特征向量基。特别地，我们证明近似流保持等谱性和李-泊松结构，且解、哈密顿量近似以及Casimir函数近似的误差仅取决于初始时刻涡量矩阵的近似精度。求解该近似模型的计算复杂度与涡量矩阵阶数呈二次方增长，若引入流函数的进一步近似则可降至线性复杂度。