A Geometric Numerical Integration of Lie-Poisson System for Ideal Compressible Isentropic Fluid

In this paper we apply a geometric integrator to the problem of Lie-Poisson system for ideal compressible isentropic fluids (ICIF) numerically. Our work is based on the decomposition of the phase space, as the semidirect product of two infinite dimensional Lie groups. We have shown that the solution of (ICIF) stays in coadjoint orbit and this result extends a similar result for matrix group discussed in [6]. By using the coadjoint action of the Lie group on the dual of its Lie algebra to advance the numerical flow, we (as in [2]) devise methods that automatically stay on the coadjoint orbit. The paper concludes with a concrete example.