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A variational finite element discretization of compressible flow. (English) Zbl 1506.65153

The paper at hand presents a novel fully discrete finite element variational integrator for compressible flows. The numerical scheme is based on the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles.
Given a triangulation of the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a subspace of the Lie algebra of this group that is shown to be isomorphic to a Raviart-Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation. The method is structure preserving and, in particular, energy conserving. Numerical experiments that illustrate convergence and conservation properties of the scheme are presented.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76M10 Finite element methods applied to problems in fluid mechanics
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
35Q35 PDEs in connection with fluid mechanics
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References:

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