SW system
Modeling and numerics
for (shallow) (water) flows
Part 1 : Saint-Venant system
E. Audusse
.
LAGA, UMR 7569, Univ. Paris 13
ANGE group (CETMEF – INRIA – UPMC)
April 2, 2013
E. Audusse
Numerics for SW flows
SW system
Shallow water system : Applications for EGRIN
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(Natural ?) hazards
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Simulations
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Equations
I
1d shallow water equations
∂t hu + ∂x
∂t h + ∂x hu = 0,
hu 2 + gh2 /2 = −gh ∂x b + Sf
with
h : water depth, b : bottom topography,
u : velocity of the water column, Sf : friction term
Saint-Venant (1871),
”Théorie du mouvement non-permanent
des eaux, avec application aux crues
des rivières et à l’introduction des marées
dans leur lit”, Comptes Rendus Acad. Sciences.
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Equations
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1d shallow water equations
∂t h + ∂x hu = 0,
∂t u + u∂x u + g ∂x η = Sf0
with
h : water depth, η : free surface,
u : velocity of the water column, Sf0 : friction term
Saint-Venant (1871),
”Théorie du mouvement non-permanent
des eaux, avec application aux crues
des rivières et à l’introduction des marées
dans leur lit”, Comptes Rendus Acad. Sciences.
E. Audusse
Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Derivation (SV1)
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Wave propagation : Lagrange formula
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Computation of the solution : Riemann invariant
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Riemann invariants
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Continuous solution for homogeneous system
∂t (u + 2c) + (u + c) ∂x (u + 2c) = 0,
∂t (u − 2c) + (u − c) ∂x (u − 2c) = 0
√
with c = gh : wave celerity
Boundary conditions
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Numerics for SW flows
SW system
Model
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Shallow water system : Derivation (SV2)
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”Physical” derivation
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Mass budget
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Numerics for SW flows
SW system
Model
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Shallow water system : Derivation (SV2)
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Gravity tem
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Pressure term
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Friction term
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Momentum budget
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Derivation (SV2)
I
Hypothesis : Almost flat bottom
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Hypothesis : Constant velocity
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Hypothesis : Rectangular channel
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Conclusion
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Numerics for SW flows
SW system
Model
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Shallow water system : Derivation (today)
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2d shallow water equations with sources
∂t h + ∇ · (hū) = 0,
gh2
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= −gh∇b −2Ω × hū −κ(h, ū)ū
∂t (hū) + ∇ · hū ⊗ ū +
2
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Derivation
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Gerbeau-Perthame [01]
Asymptotic expansion + Vertical integration of NS equations
Ferrari-Saleri [04], Marche [07], Decoene [08], Frings [12]
Topography, 2d, Coriolis, Friction, Wind, Capillarity...
Hypothesis
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Shallow water assumption
hydrostatic hypothesis
Scaled viscosity and friction
almost constant velocity
Almost flat bottom and free surface
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Numerics for SW flows
SW system
Model
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Shallow water system : Properties
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2d shallow water equations with sources
∂t h + ∇ · (hū) = 0,
gh2
∂t (hū) + ∇ · hū ⊗ ū +
I
= −gh∇b −2Ω × hū −κ(h, ū)ū
2
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Properties
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Conservation law
Hyperbolic system (wave propagation, weak solution)
Positivity of water depth (invariant domain, dry zones)
Energy (entropy) (in)equality
Non-trivial steady states
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Hyperbolicity
I
Shallow water system
∂t h + ∂x (hū) = 0
gh2
∂t (hū) + ∂x (hū 2 +
) = −gh∂x b
2
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Quasilinear form
∂t W + A(W )∂x W = S(W )
h
0
1
W =
, A(W ) =
hū
−gh + ū 2 2ū
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Hyperbolicity :
The system is hyperbolic iff matrix A is diagonalizable in R
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Numerics for SW flows
SW system
Model
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Shallow water system : Hyperbolicity (no source)
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Eigenvalues of shallow water system
p
λ± = ū ± gh
SW system is stricly hyperbolic away from the vacuum
(Influence on the time/space discretization)
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Physical meaning
Introducing the Froude number
|ū|
Fr = √
gh
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Fr < 1
Fluvial flow : Information goes in both directions
Fr > 1
Torrential flow : Information only goes downstream
(Influence on the prescribed boundary conditions)
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Numerics for SW flows
SW system
Model
Numerics
Shallow water system : Hyperbolicity
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Numerics for SW flows
SW system
Model
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Shallow water system : Hyperbolicity (with source)
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Nonconservative form of the system
∂t h + ∂x (hū) = 0
gh2
)+gh∂x b = 0
∂t (hū) + ∂x (hū 2 +
2
∂t b = 0
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Eigenvalues
λ± = ū ±
p
gh,
λ0 = 0
Not always hyperbolic...
(Possible numerical difficulties at resonant points)
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Numerics for SW flows
SW system
Model
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Shallow water system : Positivity / Energy
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Convex invariant domain
h≥0
(hū) ∈ Rd
(d = 2, 3)
.
(Possible numerical difficulties at wet/dry interfaces)
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Energy inequality
h
u2
+g
+b ,
E=
2
2
gh2
∂t (hE ) + ∂x (ū(hE +
) ≤0
2
(Can be used to ensure some stability at the discrete level)
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Numerics for SW flows
SW system
Model
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Shallow water system : Stationary states
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Continuous cases : Momentum and Head
hū
=0
∂x
ū 2
2 + g (h + z)
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Hydraulic jumps : RH relations
"
#
hū
=0
2
hū 2 + gh2
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Two dimensional case
Much more complicate...
Lake at rest
h + z = η0 ,
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ū = 0
(Has to be preserved at the discrete level)
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Numerics for SW flows
SW system
Model
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Shallow water system : Stationary states
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Flow over an obstacle
Fluvial flow
Torrential flow 1
In laboratory
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Torrential flow 2
In river
Numerics for SW flows
SW system
Model
Numerics
Numerics : Finite Volume Method
∂t u + ∇ · f (u) = 0,
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f : Rp → R2×p
Well-adapted to
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u ∈ Rp ,
Conservation properties
Discontinuous solutions
Two main type of meshes
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Structured grids
Unstructured meshes
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Primal cells : Unknowns on the triangles
Dual cells : Unknowns ”at the nodes”
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Numerics for SW flows
SW system
Model
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Numerics : Finite Volume Method
∂t u + ∇ · f (u) = 0,
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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
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Z
tn
Numerics for SW flows
SW system
Model
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Numerics : CFL condition
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Explicit scheme : Limitation on the time step
Courant, R.; Friedrichs, K.; Lewy, H. (1928), ӆber die partiellen
Differenzengleichungen der mathematischen Physik” (in German),
Mathematische Annalen 100 (1): 32-74 (English Version, 1967)
∆t n ≤
R. Courant
∆x
|max(λni )|
K.O. Friedrichs
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P. Lax
Numerics for SW flows
SW system
Model
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Numerics : CFL condition
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Numerics for SW flows
SW system
Model
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Numerics : CFL condition
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Explicit scheme : Limitation on the time step
Courant, R.; Friedrichs, K.; Lewy, H. (1928), ӆber die partiellen
Differenzengleichungen der mathematischen Physik” (in German),
Mathematische Annalen 100 (1): 32-74 (English Version, 1967)
∆t n ≤
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Physical interpretation
∆x
|max(λni )|
n
n
Fi+1/2
= F (Uin , Ui+1
)
Information at point xi+1/2
at time t ∈ [t n , t n+1 ]
has to come
from cells Ci or Ci+1 ,
not from cells Ci−1 or Ci+2
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Numerics for SW flows
SW system
Model
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Numerics : Boundary conditions
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Most common cases for SW flows
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Influence of the Froude number
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Wall
Inflow/Outflow boundaries
Fluvial in- or out- flow : One prescribed value
Torrential in (resp. out) flow : Two (resp. no) prescribed value
Possibly varying in time
Numerics
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Introduction of ghost cells
Neumann condition on missing data or on Riemann invariant
High order accuracy, source terms...
Bristeau-Coussin [INRIA Report 4282, 2001].
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Numerics for SW flows
SW system
Model
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Numerics : Center flux
F (U, V ) =
f (U) + f (V )
2
Center flux for transport equation
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Numerics for SW flows
SW system
Model
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Numerics : Properties of the solver
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Consistency
F (U, U) = f (U)
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Necessary for convergence (Lax-Wendroff theorem)
Non linear stability : scalar case
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Monotonicity : Convergence to the entropy solution
TVD and L∞ stability : Convergence up to a subsequence
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Numerics for SW flows
SW system
Model
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Numerics : Properties of the solver
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Consistency
F (U, U) = f (U)
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Necessary for convergence (Lax-Wendroff theorem)
Non linear stability : SW system
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Convex invariant domain (h ≥ 0)
Discrete in-cell entropy (energy) inequality
n
n
E (Uin+1 ) − E (Uin ) + σin G (Uin , Ui+1
) − G (Ui−1
, Uin ) ≤ 0
consistent with
∂t (hE ) + ∂x
gh2
) ≤0
(ū(hE +
2
Physical Requirements
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Numerics for SW flows
Model
Numerics
SW system
Numerics : Godunov scheme
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Solve the Riemann problem
R
u[u
(t, x) solution of the system with initial data
l ,ur ]
u(0, x) =
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ul
ur
if x ≤ 0
if x > 0
Much more easy to solve than IBVProblem !
Not so easy for complex systems...
Project the solution onto constant states
Z
Z
n+1
R
n
R
n
=
u[U n ,U n ] (∆t , x)dx +
u[U
Ui
n ,U n ] (∆t , x)dx
Ci−
i−1
i
Ci+
i
i+1
Godunov, S. K. (1959), ”A Difference Scheme for Numerical Solution of
Discontinuous Solution of Hydrodynamic Equations”, Math. Sbornik, 47,
271–306 (English version, 1969)
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Numerics for SW flows
SW system
Model
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Numerics : Godunov scheme
Solution of a Riemann problem for the Saint-Venant system
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Numerics for SW flows
SW system
Model
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Numerics : Godunov scheme
RIemann invariants for a 2 × 2 system
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Numerics for SW flows
SW system
Model
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Numerics : Godunov scheme
Rankine-Hugoniot relations for a 2 × 2 system
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Numerics for SW flows
SW system
Model
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Numerics : Godunov scheme
Some references...
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Numerics for SW flows
Model
Numerics
SW system
Numerics : Godunov scheme
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Solve the Riemann problem
R
u[u
(t, x) solution of the system with initial data
l ,ur ]
u(0, x) =
if x ≤ 0
if x > 0
ul
ur
Much more easy to solve than IBVProblem !
Not so easy for complex systems...
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Alternative (but equivalent !) way : Definition of a flux
n
n
)=
Fi+1/2
= F (Uin , Ui+1
Z
0
∆t n
R
f (u[U
n ,U n ] (t, 0))dt
i
Valid only for conservative systems
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Numerics for SW flows
i+1
SW system
Model
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Numerics : Approximate Riemann Solvers
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Basic idea
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Some numerical schemes
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To replace the exact solution by an approximate one
To ensure some conservativity and consistency properties
To use this approximate sol. to construct a numerical scheme
Roe solver
Rusanov flux
HLL, HLLC flux
Kinetic scheme
Suliciu relaxation solver
Some references
Godlewski-Raviart [96], Toro [99], Leveque [02], Bouchut [04]
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Numerics for SW flows
SW system
Model
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Numerics : HLL scheme
Approximate solution for the Saint-Venant system
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Numerics for SW flows
SW system
Model
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Numerics : HLL scheme
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Intermediate state
U∗ =
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cr Ur − cl Ul − (f (Ur ) − f (Ul )
cr − cl
Numerical flux

f (Ul )







cr f (Ul ) − cl f (Ur ) + cr cl (Ur − Ul )
F (Ul , Ur ) =

cr − cl






f (Ur )
0 < cl
cl < 0 < cr
cr < 0
Source term
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Numerics for SW flows
SW system
Model
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Source terms : Generality
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Basic definitions - with V = (U, z)
n
n
Uin+1 = Uin − σin Fl (Vin , Vi+1
) − Fr (Vi−1
, Vin )
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Consistency with the flux
Fl (V , V ) = Fr (V , V ) = F (U)
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Consistency with the sources
Fr (Vl , Vr )−Fr (Vl , Vr ) = −gh(zr −zl )+o(zr −zl ),
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Vr , Vl → V
Non linear stability : Positivity & Discrete energy inequality
Centered discretization of source terms
0
n
n
n
n
Fl (Vi , Vi+1 ) = F (Ui , Ui+1 ) +
ghin /2 ∗ (zi+1 − zi )
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Numerics for SW flows
SW system
Model
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Source terms : Well-balanced schemes
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Preservation of specific stationary state
hin + zi = η0 ,
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uin = 0
Inaccuracy of centered discretization of the sources
n
n
F h (Uin , Ui+1
) − F h (Ui−1
, Uin ) 6= 0
Test
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Positive and well-balanced numerical schemes
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Extended Godunov scheme (Chinnayya-Leroux-Seguin)
Kinetic interpretation of source term (Perth.-Sim.,ABSM)
Extended Suliciu relaxation scheme v1 (Bouchut)
Extended Suliciu relaxation scheme v2 (Galice,ACPU)
Hydrostatic reconstruction (ABBKP, Liang-Marche)
Path-conservative scheme (Castro-Macias-Pares)
Hydrostatic upwind scheme (Berthon-Foucher)
Central scheme (Kurganov, Kurganov-Noelle...)
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Numerics for SW flows
SW system
Model
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Source terms : Hydrostatic reconstruction
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Problem
Naive treatment of source terms does not preserve equilibria
Numerical oscillations
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Objective :
To extend any homogeneous solver to pbs with source terms
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To preserve stationary states of the system
To preserve stability properties of the solver
Key point :
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To introduce reconstructed variables that are in equilibrium
eq
eq
F h (Ũieq , Ũi+1
) − F h (Ũi−1
, Ũieq ) = 0
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To compute the deviation from this equilibrium variables
(Botta-Klein-Langenberg-Lützenkirchen [JCP, 04])
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Numerics for SW flows
SW system
Model
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Source terms : Hydrostatic reconstruction
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Lake at rest equilibrium
h + z = Cst,
u=0
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Reconstructed equilibrium water heights
eq
eq
hi+1/2,−
= hi + zi − zi+1/2 + , hi+1/2,+
= hi+1 + zi+1 − zi+1/2 +
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Computation of the fluxes
n
Fl (Vin , Vi+1
)
=
eq
eq
F (Ui+1/2,−
, Ui+1/2,+
)+
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!
0
g
eq
hi+1/2,−
+hi
(zi+1/2
2
Numerics for SW flows
− zi )
SW system
Model
Numerics
Source terms : Hydrostatic reconstruction
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Consistency :
The scheme is consistent with the equations
Stationary states :
The scheme preserves the discrete lake at rest equilibrium
Stability properties :
zi+1/2 = max(zi , zi+1 )
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If the homogeneous solver is positive,
the adapted solver is positive
If the homogeneous solver satisfies a discrete entropy
inequality,
the adapted scheme satisfies a semi-discrete entropy inequality
ABBKP [JCP, 05]
Numerical simulations
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Numerics for SW flows
SW system
Model
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Applications : Dam break in Malpasset
Before / After
Initial data and topography
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Numerics for SW flows
SW system
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Numerics for SW flows
Application : Tsunami
SW system
Model
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Actual problems
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Non hydrostatic SW models : Boussinesq type models
(Dutykh-Diaz [07], Bonneton et al. [11], Sainte-Marie [11]...)
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Hydrostatic NS equations : Multilayer models
(ABPS [11], ABPSM [11], Rambaud [12]...)
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Numerics for SW flows
SW system
Model
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Actual problems
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Shallow fluids : Generalized topography, Alternative rheology
(Savage-Hutter [91], Mangeney et al. [03],
Bouchut-Westdickenberg [04], Nieto et al. [10]...)
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Coupling phenomena : Erosion processes
(Hudson et al. [05], Castro et al. [08], Benkhaldoun et al.
[09, ]Cordier et al. [11], ABCDGGJSS [11]...)
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Numerics for SW flows
SW system
Model
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Applications : Evolution d’un glacier
Author : Guillaume Jouvet (FU Berlin)
mpe2013.org/fr
http://imaginary.org/exhibition/mathematics-of-planet-earth
”Un jour, une brève”
mpt.2013.fr
E. Audusse
Numerics for SW flows