Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Modeling and numerics
for (shallow) (water) flows
Part 2 : Beyond the Saint-Venant system
E. Audusse
.
LAGA, UMR 7569, Univ. Paris 13
ANGE group (CETMEF – INRIA – UPMC)
April 3, 2013
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Beginning of the story
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Saint-Venant system with sources
∂t hu + ∂x
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Numerics: Finite volume method
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∂t h + ∂x hu = 0,
hu 2 + gh2 /2 = −gh ∂x b
Positive (and entropic) scheme
Well-balanced scheme
Softwares
MASCARET, TELEMAC 2D (EDF-LNHE)
http://www.opentelemac.org/index.php/presentation?id=17
FULLSWOF (Univ. Orléans)
http://www.univ-orleans.fr/mapmo/soft/FullSWOF/
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
A discriminant test case ?
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Joint work with
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ANGE group
J. Sainte-Marie, M.O. Bristeau,
Y. Penel, P. Ung, N. Seguin, E. Godlewski
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EDF LNHE – Saint-Venant Lab.
N. Goutal, M. Jodeau, S. Boyaval
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Academic partners
F. Bouchut, C. Berthon, C. Chalons, O. Delestre, S. Cordier...
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Finite Volume Method
∂t u + ∇ · f (u) = 0,
I
u ∈ Rp ,
f : Rp → R2×p
Integration on the prism Ci × [t n , t n+1 ]
X
Uin+1 = Uin −
σijn F (Uin , Ujn , nij )
j∈V (i)
with
σijn
∆t n |Γij |
=
,
|Ci |
F (Uin , Ujn , nij )
Z
t n+1
≈
f (U(t, s)).nij dtds
Γij
E. Audusse
Z
tn
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Relaxation solver
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Godunov scheme
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Relaxation models : Introduction of a larger system
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Consistency, Stability
Complexity : Solution of the Riemann problem
that is hyperbolic (with LD fields)
that formally approaches the original ones
that ensures some stability property
for which the (homogeneous) Riemann problem is easy to solve
Numerical algorithm
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Definition of auxiliary variables from physical ones
Solution of the Riemann problem
Computation of the physical variables at the next time step
(Chen et al. [94], Jin-Xin [95], Bouchut [04]...)
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Relaxation solver
I
Example for scalar conservation law
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Original SCL
∂t u + ∂x f (u) = 0
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Relaxation model
∂t u + ∂x v
2
∂t v + c ∂x u
I
=
=
0
1
(f (u) − v )
Stability requirement
c ≥ |f 0 (u)|
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One retrieves the Rusanov scheme
(i.e. HLL scheme with cl = −cr )
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : No source
I
Suliciu relaxation model for SW system
∂t h + ∂x hū = 0
∂t hū + ∂x hū 2 + π = 0
2
∂t hπ + ∂x (hπū) + c ∂x ū =
I
gh2
−π
2
Stability criterion
c ≥h
I
1
p
gh
Eigenvalues (distincts and associated to LD fields)
c
λ± = u ± ,
h
λu = u
(Suliciu [90,92])
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : No source
I
Initialization : Relaxation source term with ” = 0”
πin = g (hin )2 /2
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Solution of the (homogeneous) Riemann problem
R
w[W
(t, x),
l ,Wr ]
E. Audusse
W= (h, ū, π)T
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : No source
I
I
Computation of the intermediate states
1
c
u ∗ = uM − ∆π,
π ∗ = πM − ∆u
2c
2
1
1
1
1
τl∗ = τl + ∆u− 2 ∆π, τr∗ = τr + ∆u+ 2 ∆π,
2c
2c
2c
2c
Waves celerities
c
λl = ul − , λu = u ∗ ,
hl
λ+ = ur +
c
,
hr
”c ≥ h
”τ =
p
gh”
Extension to vacuum : Bouchut,
”Nonlinear Stability of FVM for HCL”,
Birkhäuser, 2004
E. Audusse
Numerics for SW flows
1
”
h
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation scheme : With source
I
First version
∂t hū + ∂x
∂t h + ∂x hū = 0
hū + π + gh∂x z = 0
2
1
∂t hπ + ∂x (hπū) + c ∂x ū =
∂t z = 0
2
I
gh2
−π
2
Eigenvalues (associated to LD fields)
c
λ± = u ± ,
h
λu = u,
λ0 = 0
(Bouchut [04])
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : With source
I
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Computation of the new solution
Non conservative system
Integration process (no flux)
Properties
Consistency, WB, Positivity, Entropy inequality
Effective computation
Eigenvalues not ordered
Multiple Riemann pbs to solve
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : With source
I
Alternative version
∂t hū + ∂x
∂t h + ∂x hū = 0
hū + π + gh∂x z̃ = 0
2
2
∂t hπ + ∂x (hπū) + c ∂x ū =
∂t z̃ + ū∂x z̃
I
=
1 gh2
−π
2
1
(z − z̃)
Eigenvalues (associated to LD fields)
c
λ± = u ± ,
h
λu = u (double)
Riemann problem is under-determined...
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : With source
I
Linearized alternative version
∂t hū + ∂x
∂t h + ∂x hū = 0
hū + π + g h̄∂x z̃ = 0
2
2
∂t hπ + ∂x (hπū) + c ∂x ū =
∂t z̃ + ū∂x z̃
I
=
1 gh2
−π
2
1
(z − z̃)
Eigenvalues (associated to LD fields)
c
λ± = u ± ,
h
E. Audusse
λu = u (double)
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : With source
I
Solution of the Riemann problem
Same structure as in the conservative case...
But different intermediate states
I
Properties : Consistency, WB, Positivity, Entropy inequality
h̄ = hM
(Introduced as a wb simple Riemann solver : Galice [02])
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Numerical test
Comparison between
Suliciu without source + Hydrostatic reconstruction
and
Suliciu with source
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Towards morphodynamics
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Morphodynamics process
Suspended- and bedload
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Towards morphodynamics
I
Saint-Venant – Exner model
∂t qw + ∂x
∂ t h + ∂ x qw
= 0,
qw2
g
+ h2
h
2
= −gh∂x b −
ρs (1 − p)∂t b + ∂x qs
I
τb
,
ρw
= 0,
Properties
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Hyperbolic for classical choices of qs (h, ū, ∂x b)
Eigenvalues hard to compute except for special choices of qs
No dynamics in the solid phase
No transport in the fluid phase
(Hudson [05], Castro [08], Benkhaldoun [09], Cordier [11]...)
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Towards morphodynamics
I
Sediment flux formula (Grass [81])
qs = Ag |ū|2 ū
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Shields parameter
θ=
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|τb |/ρw
,
g (s − 1)dm
s = ρs /ρw
Threshold sediment flux formulae
q
3
qs = Φ g (s − 1)dm
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I
I
3/2
Meyer-Peter & Müller [48] : Φ = 8 (θ − θc )+
1/2
Engelund & Fredsoe [76]: Φ = 18.74(θ − θc )+ (θ1/2 − 0.7θc )
1/2
Nielsen [92] : Φ = 12 θ (θ − θc )+ ...
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Numerical strategies for bedload transport
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Steady state strategy
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External coupling (time splitting)
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Hydrodynamic computation on fixed topography
Steady state
Evolution of topography forced by hydrodynamic steady state
Efficient for phenomena involving very different time scales
Use of two different softwares for hydro- and morphodynamics
Allow to use existing solvers and different numerical strategies
Actual strategy at EDF (MASCARET-COURLIS)
Efficient for low coupling
Internal coupling
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Solution of the whole system at once
Need for a new solver
Efficient for general coupling
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Dam break : Failure for external coupling
External coupling (M. Jodeau)
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Towards morphodynamics
I
Relaxation model
∂t (hū) + ∂x
∂t h + ∂x hū = 0
hū + π + gh∂x b = 0
2
1 gh2
α2
∂x u =
−π
∂t π + ū∂x π +
h
2
∂t b + ∂x ω = 0
2
β
1
∂t ω +
− ū 2 ∂x b + 2ū∂x ω =
(qs − ω)
2
h
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Definitions
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(π, ω) : Auxiliary variables (fluid pressure, sediment flux)
> 0 : (Small) relaxation parameter
(α, β) > 0 : Have to be fixed to ensure stability
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Suliciu relaxation solver : Towards morphodynamics
I
General properties
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Formally tends to SW-Exner model when tends to 0
Always hyperbolic (h 6= 0) and LD fields
Stability requirement
a2 ≥ h2 p 0 (h) = gh3 ,
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b 2 ≥ (hu)2 + gh2 ∂u Qs
Specificities for morphodynamics
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No explicit dependency on sediment flux qS
Ordered eigenvalues
u−
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b
a
a
b
<u−
<u<u+
<u+
H
H
H
H
Riemann problem easy to solve for a < b...
But case b < a leads to unafordable computations...
With linearized version : Riemann pb may have no solution !
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
Dam break : Internal coupling
zb+Hf [m]
Free surface at t=1 s - J=4000
2
1.5
1
0.5
0
zb+Hf at t=1 s
0
2
4
6
8
10
x [m]
zb [m]
Bottom topography at t=1 s - J=4000
0.15
0.1
0.05
0
-0.05
-0.1
zb+Hf at t=1 s
0
2
4
6
8
x [m]
Internal coupling (O. Delestre)
E. Audusse
Numerics for SW flows
10
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
”Steady” flow over a movable bump
Dunes
Antidunes
Fluvial flow
Torrential flow
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Saint-Venant system
Saint-Venant – Exner system
”Steady” flow over a movable bump in 2d
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Fluid models for geophysic flows
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Incompressible Navier Stokes equations
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Efficient model for a large class of flows
CPU time (3d computations)
Complex implementation
(moving boundaries, not connected domains...)
Robustness of the softwares for stiff flows
(dam break, hydraulic jumps...)
Shallow water equations
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Easy implementation (fixed domain)
CPU time (2d computations)
Robustness of the software
Agreement with experiments... but not for all flows !
Constant velocity along the z-direction
Hydrostatic approximation
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Fluid models for geophysic flows
I
Incompressible Navier Stokes equations
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I
I
I
I
Efficient model for a large class of flows
CPU time (3d computations)
Complex implementation
(moving boundaries, not connected domains...)
Robustness of the softwares for stiff flows
(dam break, hydraulic jumps...)
Shallow water equations
I
I
I
I
I
I
Easy implementation (fixed domain)
CPU time (2d computations)
Robustness of the software
Agreement with experiments... but not for all flows !
Constant velocity along the z-direction
Hydrostatic approximation
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
From hydrostatic NS equations to multilayer models
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General Idea
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Derivation
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To derive a 2d (of hyperbolic type) model that takes into
account vertical effects in the flow
Vertical discretization of the fluid into M layers
Vertical integration of hydrostatic incomp. NS equ. by layer
Consequences
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Each layer has its own velocity
Coupling between the layers through
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Pressure term (global coupling)
Mass exchange
(”local” coupling : interface condition for mass equation)
Viscous effect
(local coupling : interface condition for momentum equation)
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Simplified case
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Hydrostatic incompressible Euler equ. with free surface
∂x u + ∂z w
= 0,
2
∂t u + ∂x u + ∂z uw + ∂x p = 0,
∂z p = −g ,
with
x ∈ R,
t > 0,
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zb (x) ≤ z ≤ η(t, x),
Boundary conditions
∂t η + uη ∂x η − wη = 0
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Vertical discretization
I
Velocity
u(t, x, z) =
M
X
1[zα−1/2 ,zα+1/2 ] ūα (t, x)
α=1
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Vertical integration
I
Equations after integration on a layer
∂t hα + ∂x hα ūα = Gα+1/2 − Gα−1/2 ,
∂t hα ūα + ∂x hα u¯2 + ghα ∂x H
α
= −ghα ∂x b + uα+1/2 Gα+1/2 − uα−1/2 Gα−1/2
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Mass exchange term
Gα+1/2 = ∂t zα+1/2 + uα+1/2 ∂x zα+1/2 − wα+1/2
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Natural conditions for lowest and uppest layers
G1/2 = GN+1/2 = 0
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What choice for the others ?
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Variable density
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Equations after integration on a layer
∂t ρα hα + ∂x ρα hα ūα = Gα+1/2 − Gα−1/2 ,
∂t ρα hα ūα + ∂xρα hα u¯2 + ∂x hα pα
α
= pα+1/2 ∂x zα+1/2 − pα−1/2 ∂x zα−1/2
+uα+1/2 Gα+1/2 − uα−1/2 Gα−1/2
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Mass exchange term
Gα+1/2 = ρα+1/2 ∂t zα+1/2 + uα+1/2 ∂x zα+1/2 − wα+1/2
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Pressure terms
Z zα+1/2
1
pα =
p(x, z, t)dz,
hα zα−1/2
E. Audusse
pα+1/2 = p(x, zα+1/2 , t),
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Different models
I
No exchange : Multifluid problems
Gα+1/2 = 0
M superposed Saint-Venant systems (Castro et al. [01]...)
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Horizontal layers
∂t zα+1/2 = ∂x zα+1/2 = 0
⇒
Gα+1/2 = −wα+1/2
1 equ. for hM and M velocity equations (Rambaud [12])
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Adaptative layers
hα (t, x) = λα H(t, x)
1 equ. for H and M velocity equations (ABPSa [11])
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Links with classical NS discretization
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Isopycnal coordinates
Stratifications are imposed in the model
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z− coordinates
Pbs to deal with complex bottom and water surface
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σ− coordinates
Well-adapted to geometry but may diffuse stratifications
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : System of equations
∂t
N
X
N
X
ρα lα H + ∂x
α=1
ρα lα Huα = 0
α=1
∂t (ρα hα uα ) + ∂x ρα hα uα2 + hα pα
= uα+1/2 Gα+1/2 − uα−1/2 Gα−1/2
+pα+1/2 ∂x zα+1/2 − pα−1/2 ∂x zα−1/2
∂t (ρα hα Tα ) + ∂x (ρα hα Tα uα )
= Tα+1/2 Gα+1/2 − Tα−1/2 Gα−1/2
ρα = ρ(Tα )
Z zα+1/2
1
pα =
p(x, z, t)dz,
hα zα−1/2
Gα+1/2 =
α
X
∂ρj hj
j=1
∂t
+
pα+1/2 = p(x, zα+1/2 , t)
α
X
∂ρj hj uj
j=1
E. Audusse
∂x
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Multilayer model : Properties
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Invariant domain
Positivity of total water depth
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Global energy inequality
N
Esv
,α =
∂
∂t
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N
X
!
N
Esv
,α
α=1
2
2
g ρα (zα+1/2
− zα−1/2
)
ρα hα uα2
+
2
2
∂
+
∂x
N
X
α=1
uα
g
N
Esv
+
ρ
h
H
α
α
,α
2
!
=0
Hyperbolicity
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Bi-layer model is hyperbolic
Possible local loss of hyperbolicity for more than three layers
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Kinetic interpretation of SW system
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Saint-Venant system
∂t hu + ∂x
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Density function
M(t, x, ξ) =
I
∂t h + ∂x hu = 0,
hu 2 + gh2 /2 = −gh ∂x b
h(t, x)
χ
c(t, x)
ξ − u(t, x)
c(t, x)
r
,
c=
gh
2
with χ an even probability function, ξ a kinetic velocity
Kinetic equation
∂t M + ξ∂x M − g ∂x z∂ξ M = Q(x, t, ξ)
Equivalency with SW system : Integration in ξ vs. (1, ξ)T
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Kinetic scheme (derivation, no source)
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Kinetic level
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I
fi
I
I
n+1
Definition of Min from (hin , uin )
Upwind scheme for the linear transport kinetic equation
=
∆t
Min −
∆x
n
fi+1/2
−
n
fi−1/2
Macroscopic scheme
Integration in ξ
Z
n
(F h )ni+1/2 = fi+1/2
dξ,
,
n
fi+1/2
=
(F q )ni+1/2 =
Z
ξMin if ξ > 0
n
ξMi+1
if ξ < 0
n
ξfi+1/2
dξ
Source term
Perthame-Simeoni [Calcolo, 01] : Numerical integration processes
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Kinetic scheme (properties, no source)
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Simplicity
Macroscopic fluxes can be computed analytically if probability
function χ is simple enough
1
χ(ω) = √ I[−√3,√3] ,
2 3
I
1/2
ω2
χ(ω) = 1 −
4 +
No need for the computation of the eigenvalues
Stability
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I
I
Positivity for water depth
Entropy inequality
CFL condition
n
∆t ≤ mini
E. Audusse
∆xi
p n
ωm
|uin | + √
ghi
2
Numerics for SW flows
!
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Multilayer system
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Extension of the kinetic interpretation
Construction of a kinetic scheme
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Modified CFL condition
Takes into account mass exchange terms
Positivity of water depth and tracer concentration
I
Extension of the hydrostatic reconstruction
Preservation of relevant equilibria
Variable density case
I
I
I
Computation of solution : Non linear problem to solve
Boussinesq assumption : Back to simple case
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : 3d dam break
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Wind-induced circulation
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Stratified fluid ?
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Lock exchange
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Gibraltar straight
E. Audusse
Numerics for SW flows
Towards morphodynamics : Suliciu relaxation
Towards 3d : Multilayer model
Derivation
Numerics : Kinetic scheme
Numerics : Ocean in bottles
.
Authors : A. Rousseau, M. Nodet, S. Minjeaud, P.O. Gaumin
(INRIA, MOISE team)
mpe2013.org/fr
http://imaginary.org/exhibition/mathematics-of-planet-earth
”Un jour, une brève”
mpt.2013.fr
E. Audusse
Numerics for SW flows