A Variational Finite Element Discretization of Compressible Flow
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A Variational Finite Element Discretization of Compressible Flow Evan S. Gawlik1 · François Gay-Balmaz2 Received: 12 October 2019 / Revised: 16 April 2020 / Accepted: 3 July 2020 © SFoCM 2020
Abstract We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on 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 certain subspace of the Lie algebra of this group. This subspace 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 that does not seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations. Keywords Structure-preserving discretization · Compressible fluid · Finite element variational integrator · Discrete diffeomorphism group · Geometric formulation of fluid dynamics Mathematics Subject Classification 65P10 · 76M60 · 37K05 · 37K65
Communicated by Douglas Arnold.
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François Gay-Balmaz [email protected] Evan S. Gawlik [email protected]
1
Department of Mathematics, University of Hawaii at Manoa, Honolulu, USA
2
LMD, CNRS and Ecole Normale Supérieure, Paris, France
123
Foundations of Computational Mathematics
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Variational Discretizations in Fluid Dynamics . . . . . . . . 2.1 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Distributional Directional Derivative and Its Properties . . . . . . 3.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . 3.2 Relation with Raviart–Thomas Finite Element Spaces . . . . . . . 3.3 The Lowest-Order Setting . . . . . . . . . . . . . . . . . . . . . . 4 The Lie Algebra-to-Vector Fields Map . . . . . . . . . . . . . . . . . 5 Finite Element Variational Integrator . . . . . . . . . . . . . . . . . . 5.1 Semidiscrete Euler–Poincaré Equations . . . . . . . . . . . . . . 5.2 The Compressible Fluid . . . . . . . . . . . . . . . . . . . . . . . 6 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Euler–Poincaré Variational Principle . . . . . . . . . . . . . . . . . . A.1 Euler–Poincaré Variational Principle for Incompressible Flows . . A.2 Euler–Poincaré Variational Principle for Compressible Flows . . . B Remarks on the Nonholonomic Euler–Poincaré Variational Formulation C Polynomials . . . . . . . . . . . . . . . . . . . . . .
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