Geometric variational finite element discretization of compressible fluids Fran¸ cois Gay-Balmaz CNRS - Ecole Normale Sup´ erieure de Paris Joint work with E. Gawlik, University of Hawaii at Manoa GDM online seminar & FoCM Workshop Geometric Integration and Computational Mechanics, June 16, 2020 Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 1 / 54
Motivation Main motivation: derivation of geometrically consistent numerical schemes for Geophysical Fluid Dynamics. Atmospheric and oceanic circulation: start with the compressible Euler equations ∂ t u + u · ∇ u + curl R × u + 1 ρ ∇ p = −∇ φ, ∂ t ρ + div( ρ u ) = 0 , ∂ t s + div( su ) = 0 . ❀ various approximations: pseudo-incompressible, anelastic, Boussinesq, shallow water, quasigeostrophic, ... Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 2 / 54
Properties : All these models (in the conservative case) admit a Hamiltonian formulation (Poisson bracket) and a Lagrangian formulation (variational principles) These approximations can be made at the level of the Lagrangian, i.e. the approximate equations can be derived geometrically from an approximate Lagrangian. All conservation laws have a geometric explanation. Main examples: Kelvin circulation theorem, conservation of potential vorticity Large scale dynamics: global behavior is more important than local high accuracy. Goal : Develop an integrator that respects as much as possible these properties. One systematic way: GEOMETRIC VARIATIONAL DISCRETIZATION . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 3 / 54
PLAN : 1. Geometric variational formulation of hydrodynamics 2. Discrete Lie group setting 3. Finite element variational integrator 4. Compressible fluids 5. Incompressible fluid with variable density 6. Some proofs 7. Connection with older approaches Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 4 / 54
1. Geometric variational formulation of hydrodynamics - Early variational principles: at least since Herivel [1955], Serrin [1959], Newcomb [1962], Lin [1963], Seliger and Whitham [1968], Bretherton [1970] - As mechanical systems on Lie groups: Arnold [1965], Marsden, Weinstein [1983], Marsden, Ratiu, Weinstein [1984], Holm, Marsden, Ratiu [1998] 1.1 Lagrangian description of hydrodynamics : Fluid dynamics in a compact manifold Ω with boundary. Lagrangian motion X ∈ Ω �→ x = ϕ ( t , X ) ∈ Ω - Configuration Lie group: G = Diff(Ω), compressible fluids; - Configuration Lie group: G = Diff vol (Ω), incompressible fluids - Lagrangian: L : TG → R , � � 1 ϕ | 2 d X − L ( ϕ, ˙ ϕ ) = 2 ̺ 0 | ˙ E ( ϕ, ∇ ϕ, ̺ 0 , S 0 ) d X Ω Ω Depends on reference fields ̺ 0 ( X ), S 0 ( X ). - Hamilton’s principle: critical action principle for the flow x = ϕ ( t , X ): � T δ L ( ϕ, ˙ ϕ ) dt = 0 , δϕ arbitrary variations − → Fluid equations in Lagr. variables . 0 Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 5 / 54
1.2 Eulerian (spatial) description of hydrodynamics Invariance of L with respect to diffeomorphisms that preserve ̺ 0 and S 0 - Eulerian fields: ϕ ◦ ϕ − 1 u := ˙ Eulerian velocity ρ := ( ̺ 0 ◦ ϕ − 1 ) | det D ϕ − 1 | Eulerian mass density s := ( S 0 ◦ ϕ − 1 ) | det D ϕ − 1 | Eulerian entropy density - Lagrangian in Eulerian description: � � 1 � 2 ρ | u | 2 − ǫ ( ρ, s ) ℓ ( u , ρ, s ) = d x Ω - Hamilton’s principle in Eulerian form (Euler-Poincar´ e): � T δ ℓ ( u , ρ, s ) dt = 0 , δ u = ∂ t ζ + [ u , ζ ] , δρ = − div( ρζ ) , δ s = − div( s ζ ) . 0 - Equations of motion: δℓ δ u = ρ ∇ δℓ δℓ δρ + s ∇ δℓ ∂ t δ u + L u δ s ∂ t ρ + div( ρ u ) = 0 , ∂ t s + div( su ) = 0 . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 6 / 54
1.3 Abstract Lie group geometric formulation Poisson structure (Lie-Poisson): Marsden, Ratiu, Weinstein [1984] Variational structure (Euler-Poincar´ e): Holm, Marsden, Ratiu [1998] G Lie group (configuration space); V vector space (advected quantities); G × V → V , ( g , a ) �→ a · g right representation; Lagrangian: L a 0 : TG → R , a 0 ∈ V , with L ( gh , ˙ gh , a 0 · h ) = L ( g , ˙ g , a 0 ) for all h ∈ G ; gg − 1 , a 0 · g − 1 ) = L ( g , ˙ Reduced Lagrangian: ℓ : g × V → R , ℓ ( u , a ) = ℓ ( ˙ g , a 0 ). g ( t ) g ( t ) − 1 ∈ g , a ( t ) = a 0 · g ( t ) − 1 ∈ V ; Given g ( t ) ∈ G , define u ( t ) = ˙ � T δ L ( g , ˙ g ) d t = 0 ⇐ ⇒ Euler-Lagrange equations 0 � T ⇐ ⇒ δ ℓ ( u , a ) d t = 0 , δ u = ∂ t v + [ v , u ] , δ a + a · v = 0 0 d δℓ δ u = δℓ δℓ δ u + ad ∗ ⇐ ⇒ δ a ⋄ a u dt Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 7 / 54
An essential modelling tool in fluid mechanics, with lots of extensions: – free boundary; – GFD; – liquid crystals; – superfluids; – fluid-structure interaction; – thermodynamics; – stochastic; – ..... e.g.: Holm [2002], FGB, Ratiu [2009], FGB, Marsden, Ratiu [2012], FGB, Ratiu, Tronci [2013], Holm [2015,....], FGB, Yoshimura [2017,....], FGB, Putkaradze [2014,....], .... Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 8 / 54
Goal : carry out the numerical discretization in a geometry preserving way by respecting the geometric variational formulation. Main idea : - “replace” this group by a finite dimensional Lie group approximation - apply the variational principles on this finite dimensional Lie group - temporal discretization in a structure preserving way – Original idea & incompressible ideal case: Pavlov, Mullen, Tong, Kanso, Marsden, Desbrun [2010] – Several developments (motivated by GFD): Rotating Boussinesq GFD equations: Desbrun, Gawlik, FGB, Zeitlin [2014] Various generalizations of discrete group: Liu, Mason, Hodgson, Tong, Desbrun [2015] Finite elements for incompressible: Natale and Cotter [2018] Anelastic and pseudo-incompressible GFD & unstructured grids: Bauer and FGB [2017] Compressible fluids & rotating shallow water: Bauer and FGB [2018] On the sphere: Brecht, Bauer, Bihlo, FGB, MacLachlan [2019] Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 9 / 54
2. Discrete Lie group setting 2.1 Discrete diffeomorphism groups T h triangulation of Ω, maximum element diameter h . Assume T h belongs to a shape-regular, quasi-uniform family {T h } : h K h max ρ K ≤ C 1 , and max h K ≤ C 2 , K ∈T h K ∈T h h K and ρ K diameter and inradius of a simplex K . Discrete functions: finite element space V h ⊂ L 2 (Ω) associated to T h Finite dimensional version of Diff(Ω): chosen as G h = { q ∈ GL ( V h ) | q 1 = 1 } , 1 discrete representative of constant function 1. Lie algebra g h = { A ∈ L ( V h , V h ) | A 1 = 0 } ❀ potential candidates to be discrete vector fields; ❀ As linear maps these discrete vector fields act as discrete derivations on V h ; ❀ Natural to choose them as discrete distributional directional derivatives. Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 10 / 54
2.2 Outline Discrete distributional directional derivatives form a subspace of the Lie algebra g h . This space is isomorphic to a well-known finite element space!! Raviart-Thomas space (main result) This space is NOT a Lie subalgebra of g h Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 11 / 54
2.3 Discrete distributional derivative H (div , Ω) = { u ∈ L 2 (Ω) n | div u ∈ L 2 (Ω) } . H 0 (div , Ω) = { u ∈ H (div , Ω) | u · n = 0 on ∂ Ω } . Definition Given u ∈ H (div , Ω), the distributional derivative in the direction u is 0 (Ω) ′ defined by ∇ dist : L 2 (Ω) → C ∞ u � � ( ∇ dist f div( gu ) d x , ∀ g ∈ C ∞ f ) g d x = − 0 (Ω) . u Ω Ω r ≥ 0 integer, T h triangulation of Ω h = { f ∈ L 2 (Ω) | f | K ∈ P r ( K ) , ∀ K ∈ T h } . V r Definition Given A ∈ gl ( V r h ) and u ∈ H 0 (div , Ω) ∩ L p (Ω) n , p > 2, we say that A approximates − u in V r h if whenever f ∈ L 2 (Ω) and f h ∈ V r h is a sequence satisfying � f − f h � L 2 (Ω) → 0, we have ∀ g ∈ C ∞ � Af h − ∇ dist f , g � → 0 , 0 (Ω) . u A is a consistent approximation of ∇ dist in V r h . u Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 12 / 54
Proposition (Gawlik and FGB) Given u ∈ H 0 (div , Ω) ∩ L p (Ω) n , p > 2, and r ≥ 0 an integer, a consistent approximation in V r h is A u ∈ gl ( V r of ∇ dist h ) given by u � � � � � A u f , g � := ( ∇ u f ) g d x − u · [ [ f ] ] { g } d s . K e K ∈T h e ∈E 0 h Considered in Natale, Cotter [2018] for the ideal fluid. Proposition For all u ∈ H 0 (div , Ω) ∩ L p (Ω) n , p > 2: A u 1 = 0 and � A u f , g � + � f , A u g � + � f , (div u ) g � = 0 ❀ well-defined linear map A : H 0 (div , Ω) ∩ L p (Ω) n → g r h ⊂ L ( V r h , V r h ) , u �→ A( u ) = A u g r h = { A ∈ L ( V r h , V r h ) | A 1 = 0 } Lie algebra of G r h . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 13 / 54
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