Numerical simulation of a depth-averaged
Euler system and comparison with analytical
solutions
ANGE TEAM : INRIA - UPMC - CNRS - CEREMA
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie
Contents
1. Introduction
2. The non hydrostatic model
3. A prediction-correction scheme
4. Results
5. Conclusion and outlook
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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1. Introduction
The inviscid shallow water model
Incompressible Euler System:
∇·u
∂u
+ (u · ∇)u + ∇p
∂t
∂E
+ ∇ · (u(E + p))
∂t
u2 + w 2
+ gz
E=
2
=
0
=
G
=
0
H = η − zb
Kinematic boundary conditions
∂η
∂η
+ us
− ws = 0
∂t
∂x
∂zb
ub
− wb = 0
∂x
u(x, t) =
1
H
Z
η
u(x, z, t)dz
zb
Boundary pressure pa = 0
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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1. Introduction
The inviscid shallow water model
Incompressible hydrostatic
Euler
∂u
∂t
∂w
|∂t
∂u
∂w
+
∂x
∂z
∂u 2
∂uw
∂p
+
+
+
∂x
∂z
∂x
∂wu
∂w 2 ∂p
+
+
+
∂x
∂z } ∂z
{z
Under the shallow water
Assumption : = hL 1
=
0
=
0
Approximation by the
average
u(x, z, t) ≈ u(x, t)
=
−g
non-hydrostatic terms
Hydrostatic Pressure
∂p
= −g
∂z
h,L: characteristic depth and
length
Saint-Venant Equation
∂Hu
∂H
+
∂t
∂x
2
∂Hu
∂
H
+
(Hu 2 + g
)
∂t
∂x
2
=
0
=
−gH
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
∂zb
∂x
4
1. Introduction
The Saint-Venant system
Many applications
•
Fluvial flow
•
Tsunami
•
Dam break ..
Figure : Wave propagation
Figure : A Tsunami
Figure : A mascaret
Figure : A dam break
Figure : An hydraulic jump
Problematic : Need more complex model to better simulate phenomena with
significant vertical acceleration and dispersion effects.
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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1. Introduction
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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2. The non hydrostatic model
A depth-averaged Euler system
Non-hydrostatic terms no more neglected
Asymptotic approximation no more required

∂ϕ
+ ∂ϕu
+ ∂ϕw


∂t
∂x
∂z
Euler system
⇒
∂p
∂ϕu
∂ϕu 2
∂ϕuw
+ ∂x + ∂z + ϕ ∂x
∂t

2

∂ϕw
ϕ = 1zb ≤z≤η
+ ∂ϕuw
+ ∂ϕw
+ ϕ ∂p
∂t
∂x
∂z
∂z
Depth Averaging
Z η
hf i =
f dz
zb
p = g(η − z) + pnh
∂
∂
hϕi +
hϕui
∂t
∂x
∂
∂
⇒
hϕui +
hϕu 2 i + hϕpi
∂t
∂x
∂
∂
hϕwi +
hϕuwi
∂t
∂x
∂
∂
hϕzi +
hϕzui
∂t
∂x
Closure relations needed
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
=
=
=
0
0
−ϕg
=
0
=
0
=
pnh |b
=
hϕwi
7
2. The non hydrostatic model
Closure relations
⇒ Energy balance:
∂
∂
hϕEi +
hϕu(E + p)i = 0
∂t
∂x
⇒ Minimization of the Energy [Levermore(1997)]
hϕu 2 i + hϕw 2 i
hϕui2 + hϕwi2
hϕEi =
+ hϕgzi ≥
+ hϕgzi
2
2
hϕui
hϕwi
ϕu =
ϕw =
hϕi
hϕi
⇒ Closure relations:
hϕu 2 i =
hϕui2
hϕi
hϕuwi =
hϕuihϕwi
hϕi
hϕzui = hϕzi
hϕui
hϕi
and,
pnh |b = 2pnh
REFERENCE:
An energy-consistent depth-averaged Euler system: derivation and properties. M-O. Bristeau et al.
Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation. J.
Sainte-Marie
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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2. The non hydrostatic model
The depth-averaged model
∂H
∂Hu
+
∂t
∂x
2
∂Hu
H
∂
+
(Hu 2 + g
+ Hpnh )
∂t
∂x
2
∂Hwu
∂Hw
+
∂t
∂x
H ∂Hu
Hu ∂(H + 2zb )
−
+
2 ∂x
2
∂x
=
0
=
−(gH + 2pnh )
=
2pnh
=
Hw
∂zb
∂x
∂
H2
∂E
+
(u(E + g
+ Hpnh )) = 0
∂t
∂x
2
•
•
•
Definition of the total pressure : p = ph + pnh , p̄h = g H2
Saint Venant system + non-hydrostatic terms
An averaged equation of the momentum equation in z gives the equation
in Hw
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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2. The non hydrostatic model
Comparison with other dispersive models
•
Green-Naghdi with zb = Cste
∂H
∂
Hu = 0
+
∂t
∂x
∂(Hu)
∂ 2 g 2
Hu + H + Hpgn = 0
+
∂t
∂x
2
∂
∂
3
(Hw) +
(Huw) = pgn
∂t
∂x
2
H ∂ ū
w̄ = −
2 ∂x
•
Energy balance
with E gn =
H
2
∂E gn
∂ +
u E gn + Hpgn = 0
∂t
∂x
g 2
2
2 2
u + 3w + 2H
See Dena Kazerani’s Poster
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
A Prediction-correction scheme type
(Chorin-Temam)

∂ 
∂t
|


 
Hu
H
∂
2
 Hu 2 + g H  +
Hu  +
2
∂x
Hw
Hwu
{z }
|
{z
} |
U
F (U)

0
∂z
∂
(Hpnh ) + pnh ∂xb 
∂x
−2pnh
{z
}
P


0
∂z
=  −gH ∂xb 
0
|
{z
}
Sb
completed with the incompressibility constraint divsw (U) = 0
We denote the flux F :


FH
F =  FHu 
FHw
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
A Prediction-correction scheme
U n+1/2 − U n
∂
+
F (U n ) = Sbn
∆t
∂x
U n+1 − U n+1/2
+ P n+1 = 0
∆t
under the constraint written at the time t n+1 :
(Hu)n+1 ∂(H n+1 + 2zb )
H n+1 ∂(Hu)n+1
.
+
divSW (U n+1 ) = −Hw n+1 −
2
∂x
2
∂x
(1)
(2)
=
(03)
Elliptic equation
Equation (3) with (2) gives:
−4H
q nh =
√
q nh
8 H
divSW (U n+1/2 )
+ Λq nh =
∂x 2
∆t
2∂
2
√
Hpnh ,
Λ = Λ(
∂ j H ∂ j zb
,
) , j = 1, 2
∂x j ∂x j
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
Prediction step
U n+1/2 − U n
∂
+
F (U n )
∆t
∂x
=
Sbn
⇒ Numerical scheme for the Saint Venant system with topography term
n+1/2
=
n
n
Hin − σ(FH,i+1/2
− FH,i−1/2
)
n+1/2
=
n
n
(Hu)ni − σ(FHu,i+1/2
− FHu,i−1/2
)
Hi
(Hu)i
Kinetic Interpretation
+
Hydrostatic reconstruction
⇒ Numerical scheme for the transport equation of Hw
n+1/2
(Hw)i
n
n
= (Hw)ni − σ(FHw,i+1/2
− FHw,i−1/2
)
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
Correction step
•
•
After computing pnh n+1
H n+1
=
H n+1/2
(Hu)n+1
=
(Hu)n+1/2 − ∆t(
(Hw)n+1
=
∂
∂zb
(Hpnh )n+1 + 2pn+1
)
nh
∂x
∂x
n+1/2
n+1
(Hw)
+ 2∆tpnh
Discretisation of the source term and of Hpnh by a centered finite
difference scheme
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
Properties of the scheme
•
Positivity of H
CFL condition of the shallow water model ⇒ Positivity of H n+1/2
Splitting scheme ⇒ H n+1 = H n+1/2
•
The lake at rest
Lake at rest ⇔ Hu = 0 and H + zb = Cste
Elliptic equation ⇒ pnh = 0
Verified by the hydrostatic part with the hydrostatic reconstruction.
⇒ We conserve properties of the Saint-Venant system
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
The elliptic equation
Elliptic equation
−4H
√
q nh
8 H
+ Λq nh =
divSW (U)
∂x 2
∆t
2∂
2
⇒ Difficulty to analyse the sign of Λ
•
Flat bottom
Λ = 16−2H
•
∂2H
∂H 2
+ 3(
)
∂x 2
∂x
Variable bottom
∂zb 2
∂ 2 zb
∂H ∂zb
∂2H
∂H 2
Λ = 16 1 + (
) −8H
+ 16
− 2H
+ 3(
)
2
∂x
∂x
∂x ∂x
∂x 2
∂x
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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3. A prediction-correction scheme
Positivity of the total pressure, Behaviour when H → 0
Hydrostatic model : Positivity of the pressure.
H
p = ph = g
2
Non-hydrostatic model : How to ensure the positivity?
H
p = ph + pnh = g + pnh
2
⇒ Solution 1
⇒ Solution 2
In the elliptic solver, not allow the
scheme to degenerate:
Transition to hydrostatic model
H = max(H, H )
−4H
∂ ρ̃Hw
∂ ρ̃Hwu
+
∂t
∂x
∂ ρ̃H
∂ ρ̃Hu
+
∂t
∂x
√
q nh
8 H
+ Λq nh =
divSW (U)
∂x 2
∆t
2∂
2
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
ρ̃M
=
2pnh
=
ρ̃M H − ρ̃H
( τ
=
1 if p > p
0 else
17
4. Results
Propagation of a solitary wave
H = H0 + asech(
x − c0 t 2
)
l
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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4. Results
Propagation of a solitary wave
Figure : H and Pnh over the time
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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4. Results
A stationary quasi-analytic solution
Stationary solution : Hu = Q0 = Cte , H = H(x)
A stationary solution of the non hydrostatic model
1.2
eta
zb
1
Q02
=
Hw
=
∂
(
+ Hp)
∂x H
=
Q0
2
Q0
2
∂w
∂x
∂
(H + 2zb )
∂x
∂zb
−(gH + 2pnh )
∂x
0.8
0.6
eta(x) , zb(x)
pnh
0.4
0.2
0
−0.2
−0.4
0
100
200
300
400
500
Space
600
700
800
900
1000
• Choosing w(x) ⇒ EDO to solve
Figure : a particular analytic solution
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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4. Results
Results - Stationary solution
Initial condition:
η = Cste
Boundary conditions:
Left : analytical debit q0
Right : analytical η
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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4. Results
Results - Comparison with hydrostatic simulation
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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4. Results
Résults - Dispersive effect
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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Conclusion and outlook
•
Properties of the scheme
-
•
Stability
Concistency
Convergence
Validation
-
Analytic solutions
Observations
•
Error approximation
•
Boundary conditions
•
2D,3D model
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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Thank you !
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie – EGRIN 2014
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