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.
Highlights
This paper develops a geometric variational discretization for compressible fluid dynamics
In classical mechanics, a time discretization of the Lagrangian variational formulation allows for the derivation of numerical schemes, called variational integrators, that are symplectic, exhibit good energy behavior, and inherit a discrete version of Noether’s theorem which guarantees the exact preservation of momenta arising from symmetries, see [25]
By means of two test cases we study whether our variational integrator is able to correctly represent the general dynamical behavior while conserving the quantities of interest discussed above
Summary
This paper develops a geometric variational discretization for compressible fluid dynamics. We develop this geometric variational discretization towards the treatment of compressible fluid dynamics This extension is based on a suitable Lie group approximation of the group of (not necessarily volume preserving) diffeomorphisms of the fluid domain, accompanied with an appropriate right invariant nonholonomic constraint obtained by requiring that Lie algebra elements are approximations of continuous vector fields. The spatial discretization of the incompressible Euler equations is obtained by applying a variational principle on the discrete diffeomorphism group Dvol(M) for an appropriate spatially discretized right invariant Lagrangian L = L(q, q) and with respect to appropriate nonholonomic constraints This approach directly follows from a variational discretization of the geometric description of the Euler equations given in [1] that we briefly mentioned above. In [6] appropriate discrete diffeomorphism groups were defined to develop variational discretization of the equations of anelastic and pseudo-incompressible fluids
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