The barotropic Navier Stokes equations

Transcription

Maillages décalés et écoulements compressibles
R. Herbin⋆ ,
with
W. Kheriji⋆† , J.-C. Latché† , B. Piar†
R. Herbin
⋆
Université de Provence
†
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
(LATP)
Paris 6, Février 2012
1 / 32
Introduction
Aim and strategy
Aim of our work
◮
Cadre : simulation numérique d’écoulements compressibles, code ISIS de l’IRSN
R. Herbin
(LATP)
2 / 32
Introduction
Aim and strategy
Aim of our work
◮
◮
But : programmer des schémas qui sont
◮
stable and précis tout Mach
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(LATP)
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Introduction
Aim and strategy
Aim of our work
◮
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◮
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suffisamment découplés pour que la résolution numérique ne soit pas trop
difficile.
méthodes de corrections de pression.
Classiques pour les fluides incompressibles (Chorin 68, Temam 69), voir
(Guermond 06) pour une synthèse.
Dévelopées aussi pour les fluides compressibles, soit en colocalisé soit en décalé..
R. Herbin
(LATP)
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Introduction
Aim and strategy
Aim of our work
◮
◮
◮
◮
difficile.
◮
On voudrait avoir des preuves théoriques de stabilité et consistance, étayés par
des résultat numériques.
R. Herbin
(LATP)
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Introduction
Aim and strategy
Aim of our work
◮
◮
◮
◮
difficile.
◮
◮
Synthèse des résultats récents : H. Kheriji Latché 2012, ESAIM Proc.
http://www.cmi.univ-mrs.fr/~ herbin
R. Herbin
(LATP)
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Introduction
Aim and strategy
Aim of our work
◮
◮
◮
◮
difficile.
◮
◮
Synthèse des résultats récents : H. Kheriji Latché 2012, ESAIM Proc.
http://www.cmi.univ-mrs.fr/~ herbin
R. Herbin
(LATP)
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Introduction
Aim and strategy
Theoretical study of discretization schemes for nonlinear problems
General picture: “Discrete non linear analysis” (see e.g. Eymard Gallouet H. 2000)
(i) Mesh dependent a priori estimates
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(LATP)
existence of the approximate solution
3 / 32
Introduction
Aim and strategy
(ii) Uniform a priori estimates
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(LATP)
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Introduction
Aim and strategy
(iii) Compactness arguments: any sequence of discrete solutions converges (weakly,
up to a subsequence) to a limit as the time and space steps tend to zero.
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(LATP)
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Introduction
Aim and strategy
(iv ) By product: a priori estimates may imply some regularity of the limit.
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(LATP)
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Introduction
Aim and strategy
(v ) Passage to the limit in (a weak formulation of) the scheme to show that the
limit is a solution of the continuous problem.
◮
For the compressible Navier-Stokes or Euler equations, no hope because of step
(ii): lack of estimates on the space translates of the unknown.
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(LATP)
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Introduction
Aim and strategy
(v ) Passage to the limit in (a weak formulation of) the scheme to show that the
limit is a solution of the continuous problem.
◮
For the compressible Navier-Stokes or Euler equations, no hope because of step
(ii): lack of estimates on the space translates of the unknown.
◮
except for the barotropic viscous Stokes equations Eymard Gallouet H. Latché
2009-2010 and steady state Navier-Stokes.
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(LATP)
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Introduction
Aim and strategy
Theoretical results for general compressible flows
◮
Discrete analogues of the continuous estimates:
◮ positivity of the density
◮ non barotropic flows: positivity of the internal energy, decrease of the total
energy
◮ barotropic flows: L2 (H1 ) estimate on the velocity
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Introduction
Aim and strategy
◮
energy
◮
Existence of a solution to the (implicit) scheme, by a topological degree
argument.
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(LATP)
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Introduction
Aim and strategy
◮
energy
◮
Existence of a solution to the (implicit) scheme, by a topological degree
argument.
◮
Consistency: If the approximate solutions converge in strong enough norms, the
limit is a weak solution to the continuous problem.
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Introduction
Space discretization
Staggered discretization
◮
Positivity of density
finite volume scheme on ρ:
ρ piecewise constant on the cells.
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(LATP)
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Introduction
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Primal mesh : M = { set of control volumes K , L, M... }.
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(LATP)
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Introduction
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◮
◮
◮
Scalar variables defined at cell centers: (pK )K ∈M , (̺K )K ∈M ,. . .
(i )
Velocity components defined at the (or some of the) edges : (vσ )σ∈F (i ) .
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(LATP)
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Introduction
◮
◮
◮
◮
◮
(i )
(i
)
Dual mesh(es) : M∗ = (Dσ )σ∈F (i ) .
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Introduction
◮
◮
◮
◮
◮
◮
(i )
(i
)
Dual mesh(es) : M∗ = (Dσ )σ∈F (i ) .
Normal velocity to the face σ denoted by vσ · nσ .
|σ |
′
K
σ
|L
σ=K
Dσ
ǫ=
Dσ |
Dσ ′
|M
K
=
L
Dσ ′
M
Figure: Primal and dual meshes for the Rannacher-Turek and Crouzeix-Raviart elements.
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(LATP)
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Introduction
The MAC mesh
: Dσ (or Kiy,j− 1 )
: Dσ (or Kix− 1 ,j )
2
2
yj+ 3
2
uyi,j+ 1
2
yj+ 1
2
uxi− 1 ,j+1
2
yj+ 1
2
yj− 1
uxi− 3 ,j
2
uxi− 1 ,j
2
uxi+ 1 ,j
2
uyi−1,j− 1
yj− 1
uyi,j− 1
2
uyi+1,j− 1
2
2
2
2
uxi− 1 ,j−1
yj− 3
2
2
xi − 1
2
uyi,j− 3
yj− 3
2
xi − 3
xi + 1
2
2
2
xi − 3
2
xi − 1
2
xi + 1
2
xi + 3
2
The dual mesh for the MAC scheme, x and y -component of the velocity.
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The continuous equations
The continuous barotropic Navier Stokes equations
∂t ρ + div(ρ u) = 0,
(mass)
∂t (ρ u) + div(ρ u ⊗ u) + ∇p − div(τ (u)) = 0,
(momentum)
ρ = Ψ(p), or p = φ(ρ)
(eos)
2µ
divu Id,
3
Ψ, φ : R+ → R+ .
τ (u) = µ(∇u + ∇t u) −
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(LATP)
µ > 0.
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Properties of barotropic Navier Stokes
Properties of the continuous equations:
◮
Positive density: for positive BC’s, ρ > 0
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(LATP)
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◮
◮
Kinetic energy balance:
1
1
∂t ( ρ |u|2 ) + div ( ρ |u|2 ) u + ∇p · u = div τ (u) · u.
2
2
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(LATP)
8 / 32
◮
◮
1
1
2
2
◮
Entropy inequality
d
dt
with H(ρ) = ρ
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(LATP)
Z
Z
1
ρ |u|2 + H(ρ) dx ≤ 0.
2
Ω
φ(ρ)
(recall that p = φ(ρ) by EOS).
ρ2
8 / 32
◮
◮
1
1
2
2
◮
Entropy inequality
d
dt
with H(ρ) = ρ
◮
Z
Z
1
ρ |u|2 + H(ρ) dx ≤ 0.
2
Ω
φ(ρ)
(recall that p = φ(ρ) by EOS).
ρ2
Recover these estimates at the discrete level...
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(LATP)
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The implicit scheme
Finite volume discretization of the mass equation
∂t ρ + div(ρ u) = 0,
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(mass)
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The implicit scheme
∂t ρ + div(ρ u) = 0,
◮
Z
(mass)
K
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(LATP)
(mass)
+ implicit time discretization
Z n+1
Z
ρ
− ρn
(ρn+1 un+1 · nK ) = 0.
+
δt
K
∂K
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The implicit scheme
∂t ρ + div(ρ u) = 0,
◮
◮
Z
(mass)
K
(mass)
Z n+1
Z
ρ
− ρn
(ρn+1 un+1 · nK ) = 0.
+
δt
K
∂K
discretization of the fluxes:
X
|K | n+1
FK ,σ = 0,
(ρK − ρnK ) +
δt
σ∈E(K )
◮
n+1
FK ,σ = |σ| ρ̌n+1
· nK ,σ , numerical flux through σ.
σ uσ
◮
ρ̌n+1
upwind approximation of ρn+1 at the face σ with respect to un+1
· nK ,σ .
σ
σ
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(LATP)
9 / 32
The implicit scheme
∂t ρ + div(ρ u) = 0,
◮
◮
Z
(mass)
K
(mass)
Z n+1
Z
ρ
− ρn
(ρn+1 un+1 · nK ) = 0.
+
δt
K
∂K
discretization of the fluxes:
X
|K | n+1
FK ,σ = 0,
(ρK − ρnK ) +
δt
σ∈E(K )
◮
n+1
FK ,σ = |σ| ρ̌n+1
· nK ,σ , numerical flux through σ.
σ uσ
◮
ρ̌n+1
upwind approximation of ρn+1 at the face σ with respect to un+1
· nK ,σ .
σ
σ
◮
Positive density: ρn+1 > 0 if (ρn > 0 and ρ > 0 at inflow boundary).
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The implicit scheme
FV-FE discretization of the momentum equation
∂t (ρ u) + div(ρ u ⊗ u) + ∇p − div(τ (u)) = 0
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The implicit scheme
◮
Z
(momentum)
Dσ
Z
Dσ
ρn+1 un+1 − ρn un+1
+
δt
Z
∂Dσ
(ρn+1 un+1 ⊗ un+1 · nK )
Z
+
(∇p n+1 − div(τ (un+1 ))) = 0.
Dσ
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(LATP)
10 / 32
The implicit scheme
◮
Z
(momentum)
Dσ
Z
Dσ
+
δt
Z
∂Dσ
(ρn+1 un+1 ⊗ un+1 · nK )
Z
+
(∇p n+1 − div(τ (un+1 ))) = 0.
Dσ
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ
(ρσ uσ − ρnσ unσ ) +
δt
ǫ∈E(Dσ )
+|Dσ |(∇p n+1 )σ − |Dσ |(divτ (un+1 ))σ = 0,
◮
X Z
p n+1 divϕ(iσ ) dx = |σ| (pLn+1 − pKn+1 ) nK ,σ · e(i ) ,
M∈M
Z
X Z
µ X
= −µ
∇un+1 · ∇ϕ(iσ ) −
div un+1 div ϕ(iσ ) .
3
K
K
|Dσ |(∇p n+1 )σ,i = −
M
◮
|Dσ |(divτ (un+1 ))σ,i
◮
R.σ
Herbin
Maillages décalés
et écoulements compressibles
i-th(LATP)
finite element shape
function.
ϕ
K ∈M
(i )
K ∈M
10 / 32
The implicit scheme
Discretization of the convection operator
◮
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
R. Herbin
(LATP)
11 / 32
The implicit scheme
◮
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
Z
Z
Discretize
+
(ρn+1 un+1 ⊗ un+1 · nK ) so as to
δt
Dσ
∂Dσ
obtain a discrete kinetic energy balance.
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
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(LATP)
11 / 32
The implicit scheme
◮
◮
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
Z
Z
Discretize
+
δt
Dσ
∂Dσ
Copy the continuous kinetic energy balance:
∂t (ρ u) + div(ρ u ⊗ u) + ∇p − div(τ (u))
=0 · u
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
1
1
∂t ( ρ |u|2 ) + div ( ρ |u|2 ) u + ∇p · u − div τ (u) · u
2
2
with some formal algebra... using ∂t ρ + div(ρ u) = 0.
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(LATP)
=0,
11 / 32
The implicit scheme
◮
◮
◮
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
Z
Z
Discretize
+
δt
Dσ
∂Dσ
=0 · u
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
1
1
2
2
Do the same at the discrete problem ?
♭ Momentum on dual cells, mass on primal cells...
R. Herbin
(LATP)
=0,
11 / 32
The implicit scheme
◮
◮
◮
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
Z
Z
Discretize
+
δt
Dσ
∂Dσ
=0 · u
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
1
1
2
2
♯ Idea: reconstruct a mass balance on the the dual cells:
X
|Dσ | n+1
n+1
∀σ ∈ Eint ,
(ρσ − ρnσ ) +
Fσ,ǫ
=0
δt
=0,
ǫ∈E(Dσ )
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(LATP)
11 / 32
The implicit scheme
◮
◮
◮
◮
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ ?
δt
ǫ∈E(Dσ )
Z
Z
Discretize
+
δt
Dσ
∂Dσ
=0 · u
n+1
Choice of ρn+1
and Fσ,ǫ
in
σ
1
1
2
2
♯ Idea: reconstruct a mass balance on the the dual cells:
X
|Dσ | n+1
n+1
∀σ ∈ Eint ,
(ρσ − ρnσ ) +
Fσ,ǫ
=0
δt
=0,
ǫ∈E(Dσ )
obtained from the primal mass balance
for
1
◮ ρσ =
|DK ,σ | ρK + |DL,σ | ρL
|Dσ |
◮ Fσ,ǫ : linear combination of the primal fluxes (F
K ,σ )σ∈E(K ) .
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(LATP)
11 / 32
The implicit scheme
Discrete convection operator
◮
Continuous convection operator
Z
Z
C(ρ, u) =
+
(ρn+1 un+1 ⊗ un+1 · nσ ).
δt
Dσ
∂Dσ
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(LATP)
12 / 32
The implicit scheme
◮
Z
Z
C(ρ, u) =
+
(ρn+1 un+1 ⊗ un+1 · nσ ).
δt
Dσ
∂Dσ
◮
Cd (ρd , ud ) =
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ
δt
ǫ∈E(Dσ )
ρσ
Fσ,ǫ
◮
◮
uǫ =
R. Herbin
1
2
(LATP)
same as in discrete dual mass balance
up
(un+1
+ un+1
σ
σ′ ) + δ
P
1
ǫ=Dσ |Dσ′ 2
n+1
n+1
sign(Fσ,ǫ
) (un+1
σ,i − uσ ′ ,i )
12 / 32
The implicit scheme
◮
Z
Z
C(ρ, u) =
+
(ρn+1 un+1 ⊗ un+1 · nσ ).
δt
Dσ
∂Dσ
◮
Cd (ρd , ud ) =
X
|Dσ | n+1 n+1
n+1 n+1
Fσ,ǫ
uǫ
δt
ǫ∈E(Dσ )
ρσ
Fσ,ǫ
◮
◮
R. Herbin
same as in discrete dual mass balance
up
(un+1
+ un+1
σ
σ′ ) + δ
(
1 upwind choice
=
0 centered choice.
uǫ =
δ up
1
2
(LATP)
P
1
ǫ=Dσ |Dσ′ 2
n+1
n+1
sign(Fσ,ǫ
) (un+1
σ,i − uσ ′ ,i ) with
12 / 32
The implicit scheme
Continuous and discrete kinetic energy balance
◮
Continuous setting: Multiply continuous momentum by u:
R. Herbin
(LATP)
·u
13 / 32
The implicit scheme
◮
·u
continuous kinetic energy balance:
1
1
∂t ( ρ |u|2 ) + div ( ρ |u|2 ) u + ∇p · u − div τ (u) · u = 0
2
2
... with some formal algebra... using ∂t ρ + div(ρ u) = 0.
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(LATP)
(♠)
13 / 32
The implicit scheme
◮
·u
1
1
2
2
◮
Discrete setting: Similarly, multiply discrete momentum by un+1
σ :
|D |
X
σ
n+1 n+1
n+1
n n
Fσ,ǫ
uǫ
(ρn+1
σ uσ − ρ σ uσ ) +
δt
(♠)
ǫ∈E(Dσ )
+ |Dσ |(∇p n+1 )σ,i − |Dσ |(divτ (un+1 ))σ = 0
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(LATP)
· uσ
13 / 32
The implicit scheme
◮
·u
1
1
2
2
◮
Discrete setting: Similarly, multiply discrete momentum by un+1
σ :
|D |
X
σ
n+1 n+1
n+1
n n
Fσ,ǫ
uǫ
(ρn+1
σ uσ − ρ σ uσ ) +
δt
(♠)
ǫ∈E(Dσ )
+ |Dσ |(∇p n+1 )σ,i − |Dσ |(divτ (un+1 ))σ = 0
discrete kinetic energy balance:
i 1
1 |Dσ | h n+1 n+1 2
ρσ (uσ ) − ρnσ (unσ )2 +
2 δt
2
· uσ
n+1 n+1 n+1
Fσ,ǫ
uσ uσ ′
ǫ=Dσ |Dσ′
n+1
un+1
= 0 with Rσn+1 ≥ 0)
σ +Rσ
X
|Dσ | n+1
n+1
...
with(LATP)
some algebra... Maillages
using décalés et écoulements
(ρσ −compressibles
ρnσ ) +
=
0.6, Février 2012
Fσ,ǫ
R. Herbin
Paris
+|Dσ | (∇p
n+1
X
n+1
)σ,i un+1
))σ
σ −|Dσ |(divτ (u
(♠)σ
13 / 32
The implicit scheme
Continuous entropy inequality
◮
1 ρ2 |u|2
;
Continuous kinetic energy equation, Ek = 12 ρ |u|2 =
2 ρ
∂t Ek + div Ek u + ∇p · u = div τ (u) · u,
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(LATP)
(♠)
14 / 32
The implicit scheme
◮
◮
1 ρ2 |u|2
;
2 ρ
(♠)
Find an entropy of the form S = Ek + H(ρ), with H satisfying
∂t H(ρ) + div H(ρ) u + p div(u) = 0.
R. Herbin
(LATP)
(♣)
14 / 32
The implicit scheme
◮
◮
1 ρ2 |u|2
;
2 ρ
(♠)
∂t H(ρ) + div H(ρ) u + p div(u) = 0.
(♣)
◮
Then, summing (♠) and (♣)
◮
∂t S + div (S + p) u − div τ (u) u = −τ (u) : ∇u.
Z
Z
d 1
d S ) dx =
[ ρ |u|2 + H(ρ)] dx ≤ 0
Integrating over Ω:
dt Ω
dt Ω 2
How to get H′ (ρ) ?
◮
R. Herbin
(LATP)
14 / 32
The implicit scheme
◮
◮
◮
◮
◮
1 ρ2 |u|2
;
2 ρ
(♠)
∂t H(ρ) + div H(ρ) u + p div(u) = 0.
(♣)
Then, summing (♠) and (♣)
∂t S + div (S + p) u − div τ (u) u = −τ (u) : ∇u.
Z
Z
d 1
d S ) dx =
[ ρ |u|2 + H(ρ)] dx ≤ 0
Integrating over Ω:
dt Ω
dt Ω 2
How to get H′ (ρ) ? Multiply mass balance ∂t ρ + div(ρ u) = 0 by H′
∂t (H(ρ)) + u · ∇(H(ρ)) + H′ (ρ)ρ divu = 0
∂t (H(ρ) + div(H(ρ)u) + (H′ (ρ)ρ − H(ρ))divu = 0
Z
φ(s)
yields (♣) for H′ (ρ)ρ − H(ρ) = p = φ(ρ), i.e. H(ρ) = ρ
.
s2
 γ
 ρ
if γ > 1,
.
For an ideal gas, = H(ρ) = γ − 1

ρ ln(ρ) if γ = 1,
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(LATP)
14 / 32
The implicit scheme
Discrete potential inequality
◮
Multiply discrete mass balance by H′ (ρK ) :
X
|K | n+1
(ρK − ρnK )H′ (ρK ) +
FK ,σ H′ (ρK ) = 0
δt
σ∈E(K )
◮
X
|K |
n+1
n
|σ| H(ρn+1
· nK ,σ
(H(ρn+1
σ )uσ
K ) − H(ρK )) +
δt
σ∈E(K )
X n+1
uσ · nK ,σ + RKn+1 = 0
+ pK
(♣)K
σ∈E(K )
H convex + upstream choice in the mass balance
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(LATP)
RKn+1 ≥ 0 ...
15 / 32
The implicit scheme
Discrete entropy inequality
◮
As in the X
continuous case,
X sum kinetic energy balance and elastic potential
(♠)σ +
(♣)K
balance:
σ∈E
X
σ∈Eint
K ∈M

h
i 1
n+1 2
n
n 2
 1 |Dσ | ρn+1
+
σ (uσ ) − ρσ (uσ )
2 δt
2
X
n+1 n+1 n+1
Fσ,ǫ
uσ uσ ′
ǫ=Dσ |Dσ′
+|Dσ | (∇p n+1 )σ un+1
− |Dσ |(divτ (un+1 ))σ un+1
σ
σ
+
X
K ∈M
i

X
X
|K
|
n+1
n

|σ| H(ρn+1
· nK ,σ + pK
un+1
· nK ,σ  ≤ 0.
(H(ρn+1
σ )uσ
σ
K ) − H(ρK )) +
δt

σ∈E(K )
σ∈E(K )
thanks to positive residual upwind term...
R. Herbin
(LATP)
16 / 32
The implicit scheme
◮
As in the X
continuous case,
(♠)σ +
(♣)K
balance:
K ∈M
σ∈E
X
σ∈Eint

h
i 1
n+1 2
n
n 2
 1 |Dσ | ρn+1
+
σ (uσ ) − ρσ (uσ )
2 δt
2
X
n+1 n+1 n+1
Fσ,ǫ
uσ uσ ′
ǫ=Dσ |Dσ′
+|Dσ | (∇p n+1 )σ un+1
σ
σ
+
X
K ∈M
◮

X
X
|K
|
n+1
n

|σ| H(ρn+1
· nK ,σ + pK
un+1
· nK ,σ  ≤ 0.
(H(ρn+1
σ )uσ
σ
K ) − H(ρK )) +
δt

σ∈E(K )
σ∈E(K )
thanks to positiveX
residual upwind term... X
X
Discrete duality:
|σ| un+1
· nK ,σ = 0
pK
|Dσ | (∇p n+1 )σ un+1
σ
σ,i +
E
◮
i
X
K ∈M
σ∈E(K )
|Dσ | (divτ (un+1 ))σ un+1
≤ 0.
σ
E
R. Herbin
(LATP)
16 / 32
The implicit scheme
◮
As in the X
continuous case,
(♠)σ +
(♣)K
balance:
K ∈M
σ∈E
X
σ∈Eint

h
i 1
n+1 2
n
n 2
 1 |Dσ | ρn+1
+
σ (uσ ) − ρσ (uσ )
2 δt
2
X
n+1 n+1 n+1
Fσ,ǫ
uσ uσ ′
ǫ=Dσ |Dσ′
+|Dσ | (∇p n+1 )σ un+1
σ
σ
+
X
K ∈M
◮

X
X
|K
|
n+1
n

|σ| H(ρn+1
· nK ,σ + pK
un+1
· nK ,σ  ≤ 0.
(H(ρn+1
σ )uσ
σ
K ) − H(ρK )) +
δt

σ∈E(K )
σ∈E(K )
thanks to positiveX
residual upwind term... X
X
Discrete duality:
|σ| un+1
· nK ,σ = 0
pK
|Dσ | (∇p n+1 )σ un+1
σ
σ,i +
E
◮
i
X
K ∈M
σ∈E(K )
|Dσ | (divτ (un+1 ))σ un+1
≤ 0.
σ
E
◮
Conservativity of the fluxes
i
X |K |
1 X |Dσ | h n+1 n+1 2
n
ρσ (uσ,i ) − ρnσ (unσ,i )2 +
(H(ρn+1
K ) − H(ρK )) ≤ 0.
2
δt
δt
R. Herbini ,E(LATP)
Maillages décalés et écoulementsKcompressibles
∈M
16 / 32
The implicit scheme
Passage to the limit : weak consistency of the scheme
◮
◮
(M(k) , δt (k) )k∈N sequence of meshes and time steps, such that h(k) and δt (k) tend to
zero as k tends to infinity.
Assume estimates : kρ(k) kT ,x,BV + kp (k) kT ,x,BV + ku (k) kT ,x,BV ≤ C , ∀k ∈ N,
ku (k) kT ,t,BV ≤ C , ∀k ∈ N.
◮
kqkT ,x,BV =
N
X
n=0
R. Herbin
(LATP)
δt
X
n
|qLn − qK
|,
kv kT ,x,BV =
σ∈E
σ=K |L
N
X
n=0
δt
X
|vσn ′ − vσn |,
ǫ∈Ē
′
σ=Dσ |Dσ
17 / 32
The implicit scheme
◮
◮
ku (k) kT ,t,BV ≤ C , ∀k ∈ N.
◮
kqkT ,x,BV =
N
X
n=0
◮
δt
X
n
|qLn − qK
|,
σ∈E
σ=K |L
Assume convergence : ρ(k) , p (k) , u (k)
3
Lr (0, T ) × Ω
R. Herbin
(LATP)
kv kT ,x,BV =
N
X
n=0
k∈N
δt
X
ǫ∈Ē
′
σ=Dσ |Dσ
3
→ (ρ̄, p̄, ū) ∈ L∞ (0, T ) × Ω , in
17 / 32
The implicit scheme
◮
◮
ku (k) kT ,t,BV ≤ C , ∀k ∈ N.
◮
kqkT ,x,BV =
N
X
n=0
◮
δt
X
n
|qLn − qK
|,
kv kT ,x,BV =
σ∈E
σ=K |L
N
X
δt
n=0
3
Lr (0, T ) × Ω
k∈N
X
ǫ∈Ē
′
σ=Dσ |Dσ
3
→ (ρ̄, p̄, ū) ∈ L∞ (0, T ) × Ω , in
Then, the limit (ρ̄, p̄, ū) satisfies
Z
Z TZ h
i
ρ(x, 0) ϕ(x, 0) dx = 0,
ρ ∂t ϕ + ρ u ∂x ϕ dx dt +
0
Z
Ω
Ω
T
0
Z h
2
i
ρ u ∂t ϕ + (ρ u + p) ∂x ϕ dx dt +
Ω
Z
ρ(x, 0) u(x, 0) ϕ(x, 0) dx = 0,
Ω
ρ = Ψ(p).
R. Herbin
(LATP)
17 / 32
The implicit scheme
◮
◮
ku (k) kT ,t,BV ≤ C , ∀k ∈ N.
◮
kqkT ,x,BV =
N
X
n=0
◮
δt
X
n
|qLn − qK
|,
kv kT ,x,BV =
σ∈E
σ=K |L
N
X
δt
n=0
3
Lr (0, T ) × Ω
k∈N
X
ǫ∈Ē
′
σ=Dσ |Dσ
3
→ (ρ̄, p̄, ū) ∈ L∞ (0, T ) × Ω , in
Then, the limit (ρ̄, p̄, ū) satisfies
Z
Z TZ h
i
ρ(x, 0) ϕ(x, 0) dx = 0,
ρ ∂t ϕ + ρ u ∂x ϕ dx dt +
0
Z
Ω
Ω
T
0
Z h
2
i
ρ u ∂t ϕ + (ρ u + p) ∂x ϕ dx dt +
Ω
Z
ρ(x, 0) u(x, 0) ϕ(x, 0) dx = 0,
Ω
ρ = Ψ(p).
and the entropy condition
Z
Z TZ
S(x, 0)ϕ(x, 0) dx ≥ 0, ∀ϕ ∈ C∞
S∂t ϕ + (S + p) u · ∇ϕ dx dt +
c Ω × [0, T ), R+ .
0
Ω
Ω
R. Herbin
(LATP)
17 / 32
The implicit scheme
Passage to the limit, sketch of proof
Proof:
◮
To get the weak formulation: classical technique (see e.g. Eymard Gallouet H.
2000): multiply the scheme by the discretization of a test function, put discrete
derivatives on the test functions and pass to the limit thanks to the estimates.
R. Herbin
(LATP)
18 / 32
The implicit scheme
Passage to the limit, sketch of proof
Proof:
◮
To get the weak formulation: classical technique (see e.g. Eymard Gallouet H.
2000): multiply the scheme by the discretization of a test function, put discrete
derivatives on the test functions and pass to the limit thanks to the estimates.
◮
Entropy Multiply (♠)σ by ϕσ , sum on σ,
Multiply (♣)K by ϕK , sum on K .
X

E∈Eint



i
1 |Dσ | h n n+1 2
1
n−1 n 2
ρσ (uσ ) − ρσ (uσ )
+
δt
2
2
X
ǫ=σ|σ′
n
n+1 n+1
Fσ,ǫ uσ
u ′
σ
n+1 ,
+|Dσ | (∇p)n+1
uσ
σ

+
X
K ∈M




h
i
|K |
H(ρn+1 ) − H(ρn
K)
K
δt
P
n+1
n+1
+ σ∈E(K ) H(ρσ ) uσ
nK ,σ
P
n+1 n
+p n+1
K ,σ
σ∈E(K ) uσ
K
= −


 ϕn

σ


 n
 ϕK , ,

X
n+1 n
Rσ
ϕσ −
E∈Eint
All terms pass to the limit, except
R. Herbin
(LATP)
P
Eint
X
K ∈M
R
n+1 n
ϕK ,
K
|Dσ |Rσn+1 ϕσ .... ≥ 0.
18 / 32
The pressure correction scheme
The pressure correction algorithm
1 - Pressure renormalization step
h1
i
h
i
1
div n ∇p̃ n+1 = div
∇p n
n
n
1/2
ρ
(ρ ρ )
(1)
2 - Prediction step
ρn ũn+1 − ρn−1 un
+ div(ρn ũn+1 ⊗ un ) + ∇p̃ n+1 − div(τ (ũn+1 )) = 0.
δt
(2)
3 - Correction step
ρn
un+1 − ũn+1
+ ∇(p n+1 − p̃ n+1 ) = 0,
δt
ρn+1 − ρn
+ div(ρn+1 un+1 ) = 0,
δt
ρn+1 = Ψ(p n+1 ).
Same stability results and passage to the limit as for the implicit scheme.
R. Herbin
(LATP)
19 / 32
Results for the Sod shock tube test
Sod shock tube test, 1D
1.2
ρ
u
=
L
1
0
,
ρ
u
=
R
0.125
0
,
EOS:p = ρ
1
n=250
n=500
n=1000
n=2000
exact
1
n=250
n=500
n=1000
n=2000
exact
0.9
0.8
0.8
pressure
velocity
0.7
0.6
0.4
0.6
0.5
0.4
0.2
0.3
0
0.2
-0.2
0.1
-2
-1
0
1
2
3
-2
-1
x
0
1
2
3
x
Sod shock tube problem – Centred scheme – MAC discretization
Velocity (left) and pressure (right).
Exact solution and numerical solution of the problem at t = 1 with CFL=1
(max velocity =1, speed of sound = 1).
Remark: residual viscosity added (either physical or by upwinding) for CR and RT
discretizations. (either physical or by upwinding) to avoid the odd-even decoupling
phenomenon.
R. Herbin
(LATP)
20 / 32
Results for the Sod shock tube test
Sod shock tube test: order of convergence
error (L1 norm)
error (L1 norm)
velocity
pressure
0.1
0.01
0.01
0.001
0.01
0.001
h
0.01
h
Sod shock tube problem – Centred scheme – L1 norm of the error between the
numerical solution and the exact solution at t = 1, as a function of the mesh (or
time) step, for CFL=1. Velocity (left) and pressure (right).
R. Herbin
(LATP)
21 / 32
The full Navier-Stokes equations
The full compressible Navier–Stokes equations
◮
balance equations
∂t ρ + div(ρ u) = 0,
(mass)
∂t (ρ u) + div(ρ u ⊗ u) + ∇p − div(τ (u)) = 0,
(momentum)
∂t (ρ E ) + div(ρ E u) + div(p u) + div(q) = div(τ (u) · u),
1
ρ = Ψ(p, e), (or p = φ(ρ, e)),
E = |u|2 + e,
2
(total energy)
◮
E : total energy, e: internal energy
◮
q = −λ∇e, heat conduction flux, λ ≥ 0.
◮
(total energy balance) - (♠) (kinetic energy balance)
(eos)
∂t (ρe) + div(ρeu) + p div(u) − div(λ∇e) = τ (u) : ∇u. (internal energy)
R. Herbin
(LATP)
22 / 32
Total energy balance and internal energy balance
◮
Positivity of e
◮ ∂t (ρe) + div(ρeu) + p div(u) − div(λ∇e) = τ (u) : ∇u. (internal energy)
◮ τ (u) : ∇u ≥ 0
◮ (mass balance)
∂t (ρe) + div(ρeu) = ρ [∂t e + u · ∇e],
◮ (EOS) ⇒ p = 0 when e = 0,
⇒ e(x, t) > 0 for all x ∈ Ω, t ∈ (0, T ).
◮
Incompressible or low Mach number flows: discretization of (internal energy)
natural choice.
◮
Using (internal energy): easy discretization which yields ed > 0.
Using (total energy) : if u = 0 and ∇q · n = 0 on ∂Ω, then:
Z
Z
(total energy)
∂t (ρ E ) + div(ρ E u) + div(p u) + div(q) =
div(τ (u) · u)
Ω
Ω
Z
1
d
ρ |u|2 + ρe dx ≤ 0,
(global total energy balance)
dt Ω 2
◮
Z
Ω
R. Herbin
(LATP)
23 / 32
Space and time discretization
Discrete total energy balance and discrete internal energy balance ?
◮
Idea: Use a “good” discretization for which we can apply the same “algebra” as
in the continuous case.
Need of a discrete kinetic energy balance equation that is compatible with the
discrete internal energy equation to recover a discrete total energy balance.
◮
Possible if:
Z
R
◮
(u)
:
∇u)
can
be
identified
to
(divτ (u))d · ud .
(τ
d
Ω
Ω
◮
RHS of (internal energy)d ≥ 0 so that ed ≥ 0.
◮
OK for FE...
a bit of work to build the MAC version of (divτ (u))d and τ (u) : ∇u)d .
◮
(H., Kheriji, Latché, 2011), (Babik, H., Kheriji, Latché, 2011)
R. Herbin
(LATP)
24 / 32
Space and time discretization
The pressure correction algorithm
1 - Pressure renormalization step
i
h1
i
h
1
∇p n
div n ∇p̃ n+1 = div
n
n
1/2
ρ
(ρ ρ )
(3)
2 - Prediction step
ρn ũn+1 − ρn−1 un
+ div(ρn ũn+1 ⊗ un ) + ∇p̃ n+1 − div(τ (ũn+1 )) = 0.
δt
(4)
3 - Correction step – Solve (simultanuously) the following non linear equations for
p n+1 , un+1 and ρn+1 :
un+1 − ũn+1
+ ∇(p n+1 − p̃ n+1 ) = 0,
δt
ρn+1 − ρn
+ div(ρn+1 un+1 ) = 0,
δt
ρn+1 e n+1 − ρn e n
+ div(ρn+1 un+1 e n+1 ) +p n+1divun+1−div(λ∇e n+1 ) = τ (un+1 ) :∇un+1 ,
δt
p n+1 = φ(ρn+1 , e n+1 ).
ρn
Same stability results and passage to the limit as for the implicit scheme.
R. Herbin
(LATP)
25 / 32
The Euler equations
Shocks !!
Euler equations:
∂t ̺ + div(̺u) = 0,
∂t (̺u) + div(̺u ⊗ u) + ∇p = 0,
∂t (̺E ) + div (̺E + p)u = 0,
p = (γ − 1) ̺e,
E =
1 2
|u| + e.
2
(total energy) 6⇔ (internal energy)
Lack of consistency in the discrete kinetic equation: for non regular solutions,
Rσ 6→ 0 with the mesh size...
R. Herbin
(LATP)
26 / 32
The Euler equations
Shocks !!
Euler equations:
∂t ̺ + div(̺u) = 0,
∂t (̺u) + div(̺u ⊗ u) + ∇p = 0,
∂t (̺E ) + div (̺E + p)u = 0,
p = (γ − 1) ̺e,
E =
1 2
|u| + e.
2
How to obtain the correct weak solutions while solving the internal energy balance ?
R. Herbin
(LATP)
26 / 32
The Euler equations
Shocks !!
Euler equations:
∂t ̺ + div(̺u) = 0,
∂t (̺u) + div(̺u ⊗ u) + ∇p = 0,
∂t (̺E ) + div (̺E + p)u = 0,
p = (γ − 1) ̺e,
E =
1 2
|u| + e.
2
How to obtain the correct weak solutions while solving the internal energy balance ?
Answer: Make the scheme ”consistent” with total energy equation. . .
counterbalance Rσ by an additional term SK in the discrete internal equation, such
that
XX
X X
δt|SK |ϕK → 0 as mesh size → 0.
δt|Rσ |ϕσ +
n
R. Herbin
σ∈E
(LATP)
n
K ∈M
26 / 32
The Euler equations
An implicit scheme
X n+1
|K | n+1
FK ,σ = 0,
(ρK − ρnK ) +
δt
σ∈E(K )
X
|Dσ | n+1 n+1
n+1 n+1
n
Fσ,ǫ
uǫ,i + |Dσ | (∇p n+1 )σ,i = 0,
(ρσ uσ,i − ρnσ uσ,i
)+
δt
ǫ∈Ē(Dσ )
X n+1 n+1
|K | n+1 n+1
FK ,σ eσ + |K | pKn+1 (div un+1 ) K = SKn+1 ,
(ρK eK − ρnK eKn ) +
δt
σ∈E(K )
pKn+1
= (γ −
1)ρn+1
K
eKn+1 .
Existence, stability
◮
If the solutions exists, and if SK ≥ 0, e ≥ 0.
◮
Kinetic energy conservation + internal energy conservation
(provided that SK may be absorbed).
◮
∃ at least a solution (topological degree argument) (and e ≥ 0.
R. Herbin
(LATP)
stability estimate
27 / 32
The Euler equations
Choice of SK
◮
(∂t (̺Ek ) + div(̺Ek u) + u · ∇p = 0, Ek = 21 |u|2 )
◮
Multiply the momentum balance equation by uσ and use the mass balance:
1 X
|Dσ |
(̺σ |uσ |2 − ̺∗σ |u∗σ |2 ) +
Fσ ′ uσ uσ ′ + (∇p)σ · uσ = −Rσ ,
2δt
2
′
ǫ=σ|σ
with:
Rσ =
i
h X
|ǫ|
|Dσ | ∗
εǫ
(uσ − uσ ′ ) ·uσ .
̺σ |uσ − u∗σ ′ |2
2δt
dǫ
′
ǫ=σ|σ
◮
Multiply kinetic egergy balance and internal energy balance by adequate
interpolates of test function ϕ
XX
n
(kinetic terms)σ ϕσ +
σ∈E
X X
n
(energy terms)K ϕK =
K ∈M
X X
n
δt|SK |ϕK +
K ∈M
The terms at the left hand side tend to what they should...
R. Herbin
XX
(LATP)
X X
XX
n
δt|Rσ |ϕσ .
σ∈E
28 / 32
The Euler equations
Choice of the source term
Rσn+1 = −
2 h
1 |Dσ | n
ρσ un+1
− unσ,i −
σ
2 δt
X
ǫ=Dσ |Dσ′
i
1 n+1
n+1
|Fσ,ǫ | (un+1
− un+1
σ
σ ′ ) uσ .
2
∀K ∈ M,
SKn+1 =
1
2
X |DK ,σ | n
2
ρK un+1
− unσ +
σ
δt
σ∈E(K )
With this choice,
XX
n
R. Herbin
σ∈E
(LATP)
δt|Rσ |ϕσ +
X X
n
X
n+1
|Fσ,ǫ
| n+1
2
(uσ ′ − un+1
σ ′′ ) .
2
ǫ∩K̄ 6=∅,
′
ǫ=Dσ
|Dσ′′
δt|SK |ϕK → 0 as mesh size → 0.
K ∈M
29 / 32
The Euler equations
Numerical tests
The test case
Test 5 Toro 2009 Chap 4.

  
ρL
5.99924
left state: uL  =  19.5975
460.894
pL

 
ρR
5.99242
right state: uR  = −6.19633
46.0950
pR

Ω = (−0.5, 0.5). Discontinuity initially located at x = 0.
Centered approximation of the velocity at the dual faces in the momentum balance.
40
35
30
25
density
(analytical)
state:
∗ intermediate
1691.65
p
=
8.68977
u∗
for x ∈ (0.028, 0.428)
20
15
10
numerical
results:
p ∈ (1691.6, 1691.8)
u ∈ (8.689, 8.690)
for x ∈ (0.032, 0.417)
R. Herbin
(LATP)
5
with
without
exact
0
-0.4
-0.2
0
x
0.2
0.4
Test 5 of [ Toro chapter 4] - Density at
t = 0.035, n = 2000 cells, with and without
corrective source terms, and analytical solution.
30 / 32
The Euler equations
Numerical tests
40
40
35
35
30
30
25
25
density
density
Density fields
20
20
15
15
10
10
5
5
n=500
n=1000
n=2000
n=500
n=1000
n=2000
0
0
-0.4
-0.2
0
0.2
0.4
-0.4
x
-0.2
0
0.2
0.4
x
Test 5 of Toro, 2009, chapter 4] - Density obtained with various meshes, with (left)
and without (right) corrective source terms. δt = h/20 CFL ≈ 1
R. Herbin
(LATP)
31 / 32
The Euler equations
Numerical tests
1800
1800
1600
1600
1400
1400
1200
1200
1000
1000
pressure
pressure
Pressure fields
800
600
600
400
200
800
400
200
n=500
n=1000
n=2000
0
n=500
n=1000
n=2000
0
-0.4
-0.2
0
0.2
0.4
-0.4
x
-0.2
0
0.2
0.4
x
Test 5 of [Toro, 2009, chapter 4] - pressure obtained with various meshes, with (left)
and without (right) corrective source terms. δt = h/20 CFL ≈ 1
R. Herbin
(LATP)
32 / 32
The Euler equations
Numerical tests
Conclusion and perspectives
◮
A suitable discretization of the convection terms allows to obtain a (local)
kinetic energy conservation identity (for colocated as well as staggered meshes).
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
The staggered scheme is stable at all Mach numbers, able to compute solutions
of inviscid flows.
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
It has been used successfully for two-phase flows .
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
Recover the entropy inequality for the Euler equations.
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
Explicit schemes: first positive results for the barotropic equations. On going
tests for Euler (Ph. D. T.T. Nguyen).
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
◮
Higher order (entropy viscosity by Guermond)
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
◮
◮
Pass to the limit in the scheme (for fixed mesh) as the Mach number tends to
0...
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
◮
◮
0...
◮
Continue the discrete non linear analysis as far as possible...
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
◮
◮
0...
◮
◮
Reactive flows
R. Herbin
(LATP)
33 / 32
The Euler equations
Numerical tests
◮
◮
of inviscid flows.
◮
To do:
◮
◮
◮
◮
0...
◮
◮
Reactive flows
R. Herbin
(LATP)
33 / 32

The barotropic Navier Stokes equations

Transcription

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