On Group-Theoretic Finite-Mode Approximation of 2D Ideal Hydrodynamics

Abstract

Structure constants of the $su(N)$ ($N$ odd) Lie algebras converge when N goes to infinity to the structure constants of the Lie algebra {\it sdiff}$(T^2)$ of the group of area-preserving diffeomorphisms of a 2D torus. Thus Zeitlin and others hypothesized that solutions of the Euler equations associated with $su(N)$ algebras converge to solutions of the Euler equations of incompressible fluid dynamics on a 2D torus. In the paper we prove the hypothesis. Our numerical experiments show the Galerkin method applied to Euler equation of hydrodynamics is computationally more efficient in the range of time in which it is stable than that based on the SU(N) approximation. However, the latter is stable for much longer time. These numerical results agree with theoretical expectations.