Abstract
This paper presents a geometric variational discretization of compressible
fluid dynamics. The numerical scheme is obtained by discretizing, in a
structure preserving way, the Lie group formulation of fluid dynamics on
diffeomorphism groups and the associated variational principles. Our framework
applies to irregular mesh discretizations in 2D and 3D. It systematically
extends work previously made for incompressible fluids to the compressible
case. We consider in detail the numerical scheme on 2D irregular simplicial
meshes and evaluate the scheme numerically for the rotating shallow water
equations. In particular, we investigate whether the scheme conserves
stationary solutions, represents well the nonlinear dynamics, and approximates
well the frequency relations of the continuous equations, while preserving
conservation laws such as mass and total energy.