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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

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Motivation

Main motivation: derivation of geometrically consistent numerical schemes for Geophysical Fluid Dynamics. Atmospheric and oceanic circulation: start with the compressible Euler equations    ∂tu + u · ∇u + curl R × u + 1 ρ∇p = −∇φ, ∂tρ + div(ρu) = 0, ∂ts + div(su) = 0. ❀ various approximations: pseudo-incompressible, anelastic, Boussinesq, shallow water, quasigeostrophic, ...

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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.

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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

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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 = Diffvol(Ω), incompressible fluids
Lagrangian: L : TG → R,

L(ϕ, ˙ ϕ) =

Ω

1 2 ̺0| ˙ ϕ|2dX −

Ω

E(ϕ, ∇ϕ, ̺0, S0)dX Depends on reference fields ̺0(X), S0(X).

Hamilton’s principle: critical action principle for the flow x = ϕ(t, X):

δ T L(ϕ, ˙ ϕ)dt = 0, δϕ arbitrary variations − → Fluid equations in Lagr. variables.

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1.2 Eulerian (spatial) description of hydrodynamics Invariance of L with respect to diffeomorphisms that preserve ̺0 and S0

Eulerian fields:

u := ˙ ϕ ◦ ϕ−1 Eulerian velocity ρ := (̺0 ◦ ϕ−1)| det Dϕ−1| Eulerian mass density s := (S0 ◦ ϕ−1)| det Dϕ−1| Eulerian entropy density

Lagrangian in Eulerian description:

ℓ(u, ρ, s) =

Ω

1 2 ρ|u|2 − ǫ(ρ, s)

dx
Hamilton’s principle in Eulerian form (Euler-Poincar´

e): δ T ℓ(u, ρ, s)dt = 0, δu = ∂tζ + [u, ζ], δρ = − div(ρζ), δs = − div(sζ).

Equations of motion:

     ∂t δℓ δu + Lu δℓ δu = ρ∇ δℓ δρ + s∇ δℓ δs ∂tρ + div(ρu) = 0, ∂ts + div(su) = 0.

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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: La0 : TG → R, a0 ∈ V , with L(gh, ˙ gh, a0 · h) = L(g, ˙ g, a0) for all h ∈ G; Reduced Lagrangian: ℓ : g × V → R, ℓ(u, a) = ℓ( ˙ gg −1, a0 · g −1) = L(g, ˙ g, a0). Given g(t) ∈ G, define u(t) = ˙ g(t)g(t)−1 ∈ g, a(t) = a0 · g(t)−1 ∈ V ; δ T L(g, ˙ g)dt = 0 ⇐ ⇒ Euler-Lagrange equations ⇐ ⇒ δ T ℓ(u, a)dt = 0, δu = ∂tv + [v, u], δa + a · v = 0 ⇐ ⇒ d dt δℓ δu + ad∗

u

δℓ δu = δℓ δa ⋄ a

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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,....], ....

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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]

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2. Discrete Lie group setting

2.1 Discrete diffeomorphism groups Th triangulation of Ω, maximum element diameter h. Assume Th belongs to a shape-regular, quasi-uniform family {Th}: max

K∈Th

hK ρK ≤ C1, and max

K∈Th

h hK ≤ C2, hK and ρK diameter and inradius of a simplex K. Discrete functions: finite element space Vh ⊂ L2(Ω) associated to Th Finite dimensional version of Diff(Ω): chosen as Gh = {q ∈ GL(Vh) | q1 = 1}, 1 discrete representative of constant function 1. Lie algebra gh = {A ∈ L(Vh, Vh) | A1 = 0} ❀ potential candidates to be discrete vector fields; ❀ As linear maps these discrete vector fields act as discrete derivations on Vh; ❀ Natural to choose them as discrete distributional directional derivatives.

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2.2 Outline Discrete distributional directional derivatives form a subspace of the Lie algebra gh. This space is isomorphic to a well-known finite element space!! Raviart-Thomas space (main result) This space is NOT a Lie subalgebra of gh

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2.3 Discrete distributional derivative H(div, Ω) = {u ∈ L2(Ω)n | div u ∈ L2(Ω)}. H0(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 ∇dist

u

: L2(Ω) → C ∞

0 (Ω)′ defined by

Ω

(∇dist

u

f )g dx = −

Ω

f div(gu) dx, ∀g ∈ C ∞

0 (Ω).

r ≥ 0 integer, Th triangulation of Ω V r

h = {f ∈ L2(Ω) | f |K ∈ Pr(K), ∀K ∈ Th}.

Definition

Given A ∈ gl(V r

h ) and u ∈ H0(div, Ω) ∩ Lp(Ω)n, p > 2, we say that A approximates −u

in V r

h if whenever f ∈ L2(Ω) and fh ∈ V r h is a sequence satisfying f − fhL2(Ω) → 0, we

have Afh − ∇dist

u

f , g → 0, ∀g ∈ C ∞

0 (Ω).

A is a consistent approximation of ∇dist

u

in V r

h . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 12 / 54

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Proposition (Gawlik and FGB)

Given u ∈ H0(div, Ω) ∩ Lp(Ω)n, p > 2, and r ≥ 0 an integer, a consistent approximation

f ∇dist

u

in V r

h is Au ∈ gl(V r h ) given by

Auf , g :=

K∈Th
K

(∇uf )g dx −

e∈E0

h

e

u · [ [f ] ]{g} ds. Considered in Natale, Cotter [2018] for the ideal fluid.

Proposition

For all u ∈ H0(div, Ω) ∩ Lp(Ω)n, p > 2: Au1 = 0 and Auf , g + f , Aug + f , (div u)g = 0 ❀ well-defined linear map A : H0(div, Ω) ∩ Lp(Ω)n → gr

h ⊂ L(V r h , V r h ),

u → A(u) = Au gr

h = {A ∈ L(V r h , V r h ) | A1 = 0} Lie algebra of G r h. Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 13 / 54

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2.4 Relation with Raviart-Thomas finite element spaces

Definition

For r ≥ 0, define Sr

h ⊂ gr h as

Sr

h := Im A = {Au ∈ L(V r h , V r h ) | u ∈ H0(div, Ω)}.

Theorem (Gawlik and FGB)

The space Sr

h ⊂ gr h, r ≥ 0, is isomorphic to the Raviart-Thomas space of order 2r

RT2r(Th) =

u ∈ H0(div, Ω) | u|K ∈ (P2r(K))n + xP2r(K), ∀K ∈ Th
.

An isomorphism is given by u ∈ RT2r(Th) → Au ∈ Sr

h.

Lie algebra elements in Sr

h ⊂ gr h correspond to discrete vector fields;

Sr

h is NOT a Lie subalgebra of gr h.

Proof: Later!

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Proposition

The kernel of A : H0(div, Ω) ∩ Lp(Ω)n → gr

h, u → A(u) = Au is

ker A = {u ∈ H0(div, Ω) ∩ Lp(Ω)n | Π2r(u) = 0} = ker Π2r, Π2r : H0(div, Ω) ∩ Lp(Ω)n → RT2r(Th) global interpolation operator. H0(div, Ω) ∩ Lp(Ω)n

Π2r

A

Sr

h

gr

h

RT2r(Th)

u ∈ H0(div, Ω), ∃! ¯

u ∈ RT2r(Th) s.t. Au = A¯

u Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 15 / 54

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2.5 The Lie algebra-to-vector fields map Construct a method valid for a large class of Lagrangians ❀ Lie algebra-to-vector fields map Use of Lagrange-d’Alembert principle of nonholonomic mechanics (e.g. Bloch [2003]) ❀ Lie algebra-to-vector fields map at least defined on Sr

h + [Sr h, Sr h]

(cannot use Au → u!!)

Definition

For r ≥ 0 define Lie algebra-to-vector field map : L(V r

h , V r h ) → [V r h ]n

A :=

n

k=1

A(I r

h(xk))ek,

I r

h : L2(Ω) → V r h : L2-orthogonal projector onto V r h . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 16 / 54

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Proposition

u ∈ H0(div, Ω) ∩ Lp(Ω): If r ≥ 1: ( Au)k = I r

h(uk),

k = 1, ..., n. If r = 0:

Au|K =

1 2|K|

e∈K
e

u · ne−(be+ − be−)ds ne− normal vector field pointing from K− to K+; be± barycenters of K±.

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H0(div, Ω) ∩ Lp(Ω)n

Π2r

A

Sr

h

gr

h

[V r

h ]n

RT2r(Th)

∆h
Rh
Not yet complete....

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3. Finite element variational integrator

3.1 Semidiscrete Euler-Poincar´ e-d’Alembert equations Given ℓ(u, ρ):

Discrete Lagrangian ℓd : gr

h × V r h → R

ℓd(A, D) := ℓ( A, D)

Action on discrete densities

g ∈ G r

h :

D · g, E = D, gE, ∀E ∈ V r

h .

B ∈ gr

h :

D · B, E = D, BE, ∀E ∈ V r

h .

Nonholonomic constraint

∆h ⊂ Sr

h ⊂ gr h Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 19 / 54

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Euler-Poincar´

e-d’Alembert variational principle Duality pairing K, A , K ∈ (gr

h)∗, A ∈ gr h.

Given g(t) ∈ G r

h define A(t) = ˙

g(t)g(t)−1 and D(t) = D0 · g(t)−1 The following are equivalent for A(t) ∈ ∆h and D(t) ∈ V r

h :

(i) δ T ℓd(A, D)dt = 0, δA = ∂tB + [B, A] and δD = −D · B, for all B(t) ∈ ∆h with B(0) = B(T) = 0. (ii)

∂t δℓd

δA , B

+
δℓd

δA , [A, B]

+

δℓd δD , D · B

= 0,

∀t ∈ (0, T), ∀ B ∈ ∆h. equivalently : ∂t δℓd δA + ad∗

A

δℓd δA − δℓd δD ⋄ D ∈ ∆◦

h,

∀t ∈ (0, T) Differential equation for D(t): ∂tD, E + D, AE = 0, ∀t ∈ (0, T), ∀ E ∈ V r

h . Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 20 / 54

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Choice of ∆h

∆h ⊂ Sr

h such that

A ∈ ∆h → δℓd δA (A, D) ∈ (gr

h)∗/∆◦ h

is a diffeomorphism for all D ∈ V r

h strictly positive.

H0(div, Ω) ∩ Lp(Ω)n

Π2r

A

Sr

h

gr

h

[V r

h ]n

RT2r(Th)

∆h
Rh
Fran¸

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4. Compressible fluids

Focus on the barotropic fluid for simplicity ℓ(u, ρ) =

Ω

1 2ρ|u|2 − ρe(ρ)

dx

All results below can be naturally extended to more general Lagrangians, such as the class of rotating stratified fully compressible fluids (Gawlik and FGB [2020]) ℓ(u, ρ, s) =

Ω

1 2ρ|u|2 + ρR · u − ǫ(ρ, s) − ρφ

dx

4.1 Discrete Lagrangian ℓd(A, D) := ℓ( A, D) =

Ω

1 2D| A|2 − De(D)

dx.

δℓd δA = I r

h(D

A)♭, δℓd δD = I r

h

1 2| A|2 − e(D) − D ∂e ∂D

.

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4.2 Choice of ∆h (Rh)

Lemma

ker

A ∈ ∆h → I r

h(D

A)♭ ∈ (gr

h)∗/∆◦ h

= {0}

⇐ ⇒ Rh ⊂ BDMr(Th) Proof. ker =

Au ∈ ∆h | I r

h(D

A)♭ ∈ ∆◦

h

=
Au ∈ ∆h | I r

h(DI r h(u)), I r h(v) = 0, ∀v ∈ Rh

=
Au ∈ ∆h | DI r

h(u), I r h(v) = 0, ∀v ∈ Rh

.

By main theorem: u ∈ Rh ⊂ RT2r(Th) ↔ Au ∈ ∆h ⊂ gr

h isomorphism

ker = {u ∈ Rh | DI r

h(u), I r h(v) = 0, ∀v ∈ Rh}.

This space is zero if and only if Rh ⊂ BDMr(Th).

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4.3 Geometric variational element scheme for compressible fluids Equations of motion     

∂t(D

A), B

+
D

A, [A, B]

+
I r

h

1 2| A|2 − e(D) − D ∂e ∂D

, D · B
= 0, ∀B ∈ ∆h

∂tD, E + D, AE = 0, ∀E ∈ V r

h .

Equivalently, in terms of ρh = D, uh = − A, σh = E, and vh = − B: Seek uh ∈ Rh and ρh ∈ V r

h such that

   ∂t(ρhuh), vh + ah(wh, uh, vh) − bh(vh, fh, ρh) = 0, ∀vh ∈ Rh ∂tρh, σh − bh(uh, σh, ρh) = 0, ∀σh ∈ V r

h , 1

2

wh = I r

h(ρhuh), fh = I r h

1

2|uh|2 − e(ρh) − ρh ∂e ∂ρh

, and
ah(w, u, v) =
K∈Th
K

w · (v · ∇u − u · ∇v) dx +

e∈E0

h

e

(v · n[u] − u · n[v]) · {w}ds,

bh(w, f , g) =
K∈Th
K

(w · ∇f )g dx −

e∈E0

h

w · [[f ]]{g}ds.

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4.4 Temporal discretization OPTION 1: variational discretization OPTION 2: energy preserving discretization

1

∆t δℓd δAk − δℓd δAk−1

, Bk
+1

2

δℓd

δAk−1 + δℓd δAk , [Ak−1/2, Bk]

+ Fk−1/2, Dk−1/2 · Bk = 0,

∀ Bk ∈ ∆h, Dk − Dk−1 ∆t , Ek

+ Dk−1/2 · Ak−1/2, Ek = 0,

∀Ek ∈ V r

h .

where Fk−1/2 = 1 2

Ak−1 ·

Ak − f (Dk−1, Dk), f (x, y) = ye(y) − xe(x) y − x (reminiscent of a discrete gradient method Hairer, Lubich, Wanner [2006])

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4.5 Convergences Rotating shallow water, Ω = (−1, 1) × (−1, 1); u(x, y, 0) = (0, 0), ρ(x, y, 0) = 2 + sin(πx/2) sin(πy/2); Th, uniform, Rh = RTr(Th), V r

h , r = 0, 1, 2.

L2-errors in u and ρ at T = 0.5 by comparing with an “exact solution” obtained with h = 2−5, r = 2. (∆t = 0.00625, h = 2−j, j = 0, 1, 2, 3)

r h−1 uh − uL2(Ω) Rate ρh − ρL2(Ω) Rate 1 3.58 · 10−1 2.10 · 10−1 2 1.84 · 10−1 0.96 1.17 · 10−1 0.85 4 9.31 · 10−2 0.99 5.58 · 10−2 1.06 8 4.64 · 10−2 1.00 2.74 · 10−2 1.03 1 1.43 · 10−1 1.00 · 10−1 1 2 4.36 · 10−2 1.71 2.43 · 10−2 2.05 4 1.37 · 10−2 1.68 6.85 · 10−3 1.83 8 4.40 · 10−3 1.63 1.74 · 10−3 1.97 1 2.78 · 10−2 1.83 · 10−2 2 2 7.80 · 10−3 1.83 4.61 · 10−3 1.99 4 1.81 · 10−3 2.11 6.35 · 10−4 2.86 8 4.50 · 10−4 2.00 1.15 · 10−4 2.46

Convergence wrt ∆t of L2-errors in u and ρ at T = 0.5 (h = 2−4, r = 2)

∆t−1 uh − uL2(Ω) Rate ρh − ρL2(Ω) Rate 2 4.93 · 10−2 9.95 · 10−2 4 1.68 · 10−2 1.55 3.12 · 10−2 1.67 8 5.03 · 10−3 1.74 8.92 · 10−3 1.81 16 1.44 · 10−3 1.80 2.43 · 10−3 1.88

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4.6 Rayleigh-Taylor instability ℓ(u, ρ, s) =

Ω

1 2ρ|u|2 − ρe(ρ, η) − ρφ

dx,

e(ρ, η) = Keη/Cv ργ−1 where γ = 5/3, K = Cv = 1, φ = −y Ω = (0, 1/4) × (0, 1), Rh = RT0(Th) and V 1

h on uniform Th, h = 2−8, with upwinding

(later), ∆t = 0.01. ρ(x, y, 0) = 1.5 − 0.5 tanh y − 0.5 0.02

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Contours of the mass density at t = 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 in the Rayleigh-Taylor instability simulation with the energy-preserving time discretization Energy preserved exactly up to roundoff errors.

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5. Fluids with variable density

5.1 Equations and variational principle

Lagrangian version (Hamilton’s principle): seek ϕ : [0, T] → Diffvol(Ω) such that

δ T L(ϕ, ˙ ϕ)dt = 0, L(ϕ, ˙ ϕ) =

Ω

1 2̺0| ˙ ϕ|2dX for all δϕ vanishing at the endpoints.

Eulerian version (Euler-Poincar´

e) δ T ℓ(u, ρ)dt = 0, ℓ(u, ρ) =

Ω

1 2ρ|u|2dx for all variations δu, δρ of the form δu = ∂tζ + [u, ζ], δρ = − div(ρζ)

Incompressible fluids with variable density

ρ(∂tu + u · ∇u) = −∇p, ∂tρ + div(ρu) = 0, div u = 0.

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5.2 Discrete setting G r

h = {q ∈ GL(V r h ) | q1 = 1, qf , qg = f , g},

gr

h = {A ∈ L(V r h , V r h ) | A1 = 0, Af , g + f , Ag = 0, ∀ f , g ∈ V r h }

Rh = {u ∈ BDMr(Th), div u = 0} ❀ ∆h ⊂ gr

h

ℓd(A, D) := ℓ( A, D) =

Ω

1 2D| A|2dx. δℓd δA = I r

h(D

A)♭, δℓd δD = I r

h

1 2| A|2 .

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5.3 Geometric variational element scheme for incompressible fluids with variable density Equations of motion     

∂t(D

A), B

+
D

A, [A, B]

+
I r

h

1 2| A|2 , D · B

= 0, ∀B ∈ ∆h

∂tD, E + D, AE = 0, ∀E ∈ V r

h .

Equivalently, in terms of ρh = D, uh = − A, σh = E, and vh = − B: Seek uh ∈ BDMr(Th), ρh ∈ V r

h , ph ∈ V r−1 h

∩ L2

=0(Ω) such that

         ∂t(ρhuh), vh + ah(wh, uh, vh) − bh(vh, fh, ρh) = ph, div vh , ∀vh ∈ BDMr(Th) ∂tρh, σh − bh(uh, σh, ρh) = 0, ∀σh ∈ V r

h , 1

2 div uh, q = 0, ∀qh ∈ V r−1

h

∩ L2

=0(Ω)

with wh = I r

h(ρhuh), fh = I r h

1

2|uh|2

.

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5.4 Spatial discretization

Proposition (Gawlik & FGB)

The semidiscrete solution satisfies d dt

Ω

ρhdx = 0 d dt

Ω

ρ2

hdx = 0

d dt

Ω

ρh|uh|2dx = 0 div uh = 0

Proof. Use a(w, u, v) = −a(w, v, u) and b(u, f , g) = −b(u, g, f )

uh ∈ BDMr(Th) ❀ div uh ∈ V r−1

h

Ω div uhdx =
∂Ω uh · nds = 0 ❀ div uh ∈ V r−1

h

∩ L2

=0(Ω)

So div uh = 0 by last equation.

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5.5 Temporal discretization Seek uk ∈ BDMr(Th), ρk ∈ V r

h , pk ∈ V r−1 h

∩ L2

=0(Ω) such that

                     ρk+1uk+1 − ρkuk ∆t , v + ah ρkuk + ρk+1uk+1 ∆t , uk + uk+1 2 , v

−bh
v, Ih(1

2uk · uk+1), ρk + ρk+1 2

= pk+1, div v , ∀v ∈ BDMr(Th)

ρk+1 − ρk ∆t , σ − bh uk + uk+1 2 , σ, ρk + ρk+1 2

= 0, ∀σh ∈ V r

h , 1

2 div uk+1, q = 0, ∀q ∈ V r−1

h

∩ L2

=0(Ω)

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SLIDE 35

Proposition (Gawlik & FGB)

The fully discrete solution of incompressible fluid with variable density satisfies

Ω

ρk+1dx =

Ω

ρkdx

Ω

ρ2

k+1dx =

Ω

ρ2

kdx

Ω

1 2ρk+1|uk+1|2dx =

Ω

1 2ρk|uk|2dx div uk = 0

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5.6 Previous works For ideal fluid (ρ = 1): recovers Guzman, Shu, Sequeria [2016] and Natale, Cotter [2018]. For incompressible fluid with variable density: Closest work is Guermond and Quartepelle [2000] Their spatial discretization preserves

Ω

ρhdx,

Ω

ρ2

hdx,

Ω

ρh|uh|2dx but incompressibility constraint is only satisfied in a weak sense. Temporal discretization does not preserve

Ω

ρ2

kdx

and

Ω

ρk|uk|2dx

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5.7 Rayleigh-Taylor instability Ω = (−1/2, 1/2) × (−2, 2) with initial conditions u(x, y, 0) = (0, 0), ρ(x, y, 0) = 2 + tanh y + 0.1 cos(2πx) 0.1

.

add a gravitational forcing term (0, −g)ρk+1/2, v, g = 10 Finite element spaces uk ∈ Rh = RT0(Th), ρk ∈ DG1(Th), and pk ∈ DG0(Th) ∩ L2

=0(Ω)
n uniform Th, h = 2−j, j = 4, 5, 6, ∆t = 0.01, with upwind.

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Figure: Density contours at t = 0.8, 0.95, 1.1, 1.25 in the Rayleigh-Taylor instability simulation with h = 2−4.

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SLIDE 39

Figure: Density contours at t = 0.8, 0.95, 1.1, 1.25 in the Rayleigh-Taylor instability simulation with h = 2−5.

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Figure: Density contours at t = 0.8, 0.95, 1.1, 1.25 in the Rayleigh-Taylor instability simulation with h = 2−6.

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Figure: Density contours at t = 0.8, 0.95, 1.1, 1.25 in the Rayleigh-Taylor instability simulation with h = 2−5 and no upwinding.

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Figure: Density contours at t = 0.8, 0.95, 1.1, 1.25 in the Rayleigh-Taylor instability simulation, obtained using Guermond and Quartapelle [2000] with h = 2−5, ∆t = 0.01.

The two methods under comparison produce qualitatively similar results for t < 1, and begin to deviate somewhat as t increases.

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0.2 0.4 0.6 0.8 1 10−17 10−13 10−9 10−5 10−1 t Squared density error h = 2−4 (upwind) h = 2−5 (upwind) h = 2−6 (upwind) h = 2−4 (no upwind) h = 2−5 (no upwind) h = 2−6 (no upwind) h = 2−4 G-Q [2000] h = 2−5 G-Q [2000] h = 2−6 G-Q [2000]

Figure: Squared density errors |1 − F(t)/F(0)|, F(t) =

Ω ρh(t)2 dx, in the RTI.

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0.2 0.4 0.6 0.8 1 10−17 10−13 10−9 10−5 10−1 t Energy error h = 2−4 (upwind) h = 2−5 (upwind) h = 2−6 (upwind) h = 2−4 (no upwind) h = 2−5 (no upwind) h = 2−6 (no upwind) h = 2−4 G-Q [2000] h = 2−5 G-Q [2000] h = 2−6 G-Q [2000]

Figure: Energy errors |1 − E(t)/E(0)|, E(t) =

Ω

1

2 ρh(t)uh(t) · uh(t) + ρh(t)gy

dx, in RTI.

(The curves labelled G-Q [2000] appear nonsmooth because the sign of 1 − E(t)/E(0) changes from negative to positive near t = 0.)

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0.2 0.4 0.6 0.8 1 10−18 10−13 10−8 10−3 102 t Velocity divergence h = 2−4 (upwind) h = 2−5 (upwind) h = 2−6 (upwind) h = 2−4 (no upwind) h = 2−5 (no upwind) h = 2−6 (no upwind) h = 2−4 G-Q [2000] h = 2−5 G-Q [2000] h = 2−6 G-Q [2000]

Figure: L2-norm of the divergence of the velocity field in the Rayleigh-Taylor instability

simulation.

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6. Proof of the theorem

Theorem (Gawlik and FGB)

The space Sr

h of discrete distributional derivatives on V r h is isomorphic to RT2r(Th):

u ∈ RT2r(Th) ← → Au ∈ Sr

h

Recall: Sr

h = {Au | u ∈ H},

H := H0(div, Ω) ∩ Lp(Ω)n Auf , g :=

K∈Th
K

(∇uf )g dx −

e∈E0

h

e

u · [ [f ] ]{g} ds RT2r(Th) =

u ∈ H0(div, Ω) | u|K ∈ (P2r(K))n + xP2r(K), ∀K ∈ Th
Basis of the dual (Brezzi, Fortin [1991])

u →

e

(u · n)p ds, p ∈ P2r(e), e ∈ E0

h,

u →

K

u · p dx p ∈ P2r−1(K)n, K ∈ Th.

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STEP 1: Working with A∗:

Linear map: A : H → gl(V r

h ),

u → A(u) = Au Dual map A∗ : gl(V r

h )∗ → H∗

We have dim(Im A) = dim(Im A∗) Im A∗ = N

i=1

ciA∗(fi ⊗ gi) ∈ H∗

N ∈ N, fi, gi ∈ V r

h , ci ∈ R, i = 1, 2, . . . , N

,

A∗(f ⊗ g)(u) = f , Aug

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STEP 2: Im A∗ is spanned by the functionals

u →

e

(u · n)pq ds, p, q ∈ Pr(e), e ∈ E0

h,

u →

K

(u · ∇q)p dx p, q ∈ Pr(K), K ∈ Th. Given e = K1 ∩ K2 choose f |K1 = p ∈ Pr(K1), f |Ω\K1 = 0 and g|K2 = −2q ∈ Pr(K2), g|Ω\K2 = 0 ❀ f , Aug =

e

(u · n)pq ds Given K choose f |K = p ∈ Pr(K), f |Ω\K = 0 and g|K = q ∈ Pr(K), g|Ω\K = 0 ❀ f , Aug =

K

(u · ∇q)p dx − 1 2

∂K

(u · n)pq ds. Appropriate linear combinations yields the desired functionals.

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STEP 3: Im A∗ is spanned by the functionals

u →

e

(u · n)p ds, p ∈ P2r(e), e ∈ E0

h,

u →

K

u · p dx p ∈ P2r−1(K)n, K ∈ Th. First functional: follows from N

i=1

piqi | N ∈ N, pi, qi ∈ Pr(K), i = 1, 2, . . . , N

= P2r(K).

Trivial Second functional: follows from N

i=1

pi∇qi | N ∈ N, pi, qi ∈ Pr(K), i = 1, 2, . . . , N

= P2r−1(K)n, n = 2, 3.

By induction.

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7. Connections with older approaches

In the lowest-order setting (r = 0), V 0

h = RN.

g0

h =

A ∈ RN×N
N

j=1 Aij = 0

the components of A = Au relative to the basis {1Ki }i ⊂ Vh are

Aij = − 1 2|Ki|

Ki ∩Kj

u · n ds, j ∈ N(i), Aii = 1 2|Ki|

Ki

div u dx, (1) and Aij = 0 for all other j. hence the nonholonomic constraint is S0

h = Im A =

A ∈ g0

h | Aij = 0, ∀j /

∈ N(i) ∪ {i}, ATΘ + ΘA is diagonal

See Bauer, FGB [2018].

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8. Conclusion

Used the geometric formulation of hydrodynamics to design structure preserving spatial and temporal discretization of fluid flow valid in 2D and 3D. We connected with finite elements, one of the largest class of discretization methods for fluids. All the steps are guided by geometry (including choice of finite element space). Interesting connection with nonholonomic mechanics. The geometric approach yields new schemes via a constructive approach and more conservation properties. This geometric framework allows for several natural extensions to

ther fluid models - under investigation.

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Continous diffeomorphism Discrete diffeomorphisms Diff(Ω) ∋ ϕ G r

h ∋ g

Lie algebra Discrete diffeomorphisms X(Ω) ∋ u gr

h ∋ A

Group action on functions Group action on discrete functions f → f ◦ ϕ f → g −1f Lie algebra action on functions Lie algebra action on discrete functions f → ∇uf f → −Af Group action on densities Group action on discrete densities ρ → (ρ ◦ ϕ)Jϕ D → g −1 · D Lie algebra action on densities Lie algebra action on discrete densities ρ → div(ρu) D → −A · D Hamilton’s principle Lagrange-d’Alembert principle δ T

0 Lρ0(ϕ, ˙

ϕ)dt = 0, δ T

0 LD0(g, ˙

g)dt = 0, ˙ gg −1 ∈ ∆h, for arbitrary variations δϕ for variations δgg −1 ∈ ∆h Eulerian velocity and density Eulerian discrete velocity and discrete density u = ˙ ϕ ◦ ϕ−1, ρ = (ρ0 ◦ ϕ−1)Jϕ−1 A = ˙ gg −1, D = g · D0 Euler-Poincar´ e principle Euler-Poincar´ e-d’Alembert principle δ T

0 ℓ(u, ρ)dt = 0, δu = ∂tζ + [ζ, u],

δ T

0 ℓ(A, D)dt = 0,

δA = ∂tB + [B, A], δρ = − div(ρζ) δD = B · D, A, B ∈ ∆h Compressible Euler equations Discrete compressible Euler equations ∂t δℓ

δu + Lu δℓ δu = ρ∇ δℓ δρ

∂t

δℓd δA + ad∗ A δℓd δA − δℓd δD ⋄ D ∈ ∆◦ h

∂tρ + div(ρu) = 0 ∂tD − A · D = 0

Fran¸ cois Gay-Balmaz (CNRS-ENS) Variational discretization of compressible fluids 52 / 54

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W. Bauer and F. Gay-Balmaz.

Towards a geometric variational discretization of compressible fluids: the rotating shallow water equations.

J. Comp. Dyn., 16(1), 1–37, 2019.
R. Brecht, W. Bauer, A. Bihlo, F. Gay-Balmaz, and S. MacLachlan

Variational integrator for the rotating shallow-water equations on the sphere.

Q. J. R. Meteorol. Soc., 145, 1070–1088, 2019.
E. S. Gawlik and F. Gay-Balmaz.

A variational finite element discretization of compressible flow. arXiv:1910.05648, 2019.

E. S. Gawlik and F. Gay-Balmaz.

A conservative finite element method for the incompressible Euler equations with variable density.

J. Comp. Phys., 412(2020) 109439.
A. Natale and C. J. Cotter.

A variational finite-element discretization approach for perfect incompressible fluids. IMA Journal of Numerical Analysis, 38(3), 1388–1419 (2018).

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun.

Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6), 443–458 (2011).

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THANK YOU!

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