SHARK-FV Arnaud Duran Shallow Water Equations - Generalities dG discretization Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives Recent advances on numerical simulation in coastal oceanography Arnaud Duran Institut Camille Jordan - Université Claude Bernard Lyon 1 Work in collaboration with F. Marche (IMAG Montpellier) SHARK-FV 2017 Conference - Ofir, May 19. Arnaud Duran (ICJ) SHARK-FV 19/05/2017 1 / 23
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SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Recent advances onnumerical simulation in coastal oceanography
Arnaud DuranInstitut Camille Jordan - Université Claude Bernard Lyon 1
Work in collaboration with F. Marche (IMAG Montpellier)
SHARK-FV 2017 Conference - Ofir, May 19.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 1 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Outline
1 Shallow Water Equations - Generalities
2 dG discretization
3 Extension to dispersive equations
4 Handling breaking waves (work with G. Richard)
5 Perspectives
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 2 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Outline
1 Shallow Water Equations - GeneralitiesIntroductionNumerical stability criteriaReformulation of the SW equations
2 dG discretization
3 Extension to dispersive equations
4 Handling breaking waves (work with G. Richard)
5 Perspectives
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 3 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Coastalhydrodynamic
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Coastalhydrodynamic
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Coastalhydrodynamic
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Coastalhydrodynamic
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Tsunamis
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Shallow Water Equations
2D Formulation
∂tU +∇.G (U) = B(U, z) .
U =
h
hu
hv
, G(U) =
hu hv
12gh2 + hu2 huv
huv12gh2 + hv2
, B(U, z) =
0−gh∂xz−gh∂yz
Application field : some examples
Rivers,dam breaks
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 4 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical issues
Stability criteriaPreservation of steady states :→ (C-property) [Bermudez & Vázquez, 1994]
h + z = cte , u = 0 .
Robustness : preservation of the water depth positivity.Entropy inequalities.
[Y. Xing, X. Zhang, 2013] Extension to triangular meshes
The methodrelies on a special quadrature rule.
Figure: Nodes locations for the special quadrature - P2 and P3
reduces to the study of a convex combination of first orderFinite Volume schemes.Arnaud Duran (ICJ) SHARK-FV 19/05/2017 10 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Outline
1 Shallow Water Equations - Generalities
2 dG discretization
3 Extension to dispersive equationsMotivationsThe physical modelReformulation of the systemHigh order derivatives
4 Handling breaking waves (work with G. Richard)
5 Perspectives
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 11 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Interest of dispersive equations
ObjectiveExtend the range of applicability of the computations at coast.. Describe the non-linearities before the breaking point.. Dispersive equations : O(µ2)-accurateShallow Water equations : O(µ)-accurate .
Shallowness parameter : µ =h2
0
λ20.
Dispersive equations Shallow Water
State of the art. 1d works : [Antunes Do Carmo et al] (FD, 1993), [Cienfuegos et al] (FV, 2006),
[Dutykh et al] (FV, 2013), [Panda et al] (dG, 2014), [AD, Marche] (dG, 2015). 2d works : [Marche, Lannes] (Hybrid FV/FD, cartésien, 2015),
[Popinet] (Hybrid FV/FD, cartésien, 2015). Unstructured meshes : Weakly non linear models (Boussinesq - type ).
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Model presentation
. [P. Bonneton et al, 2011] 1d derivation and optimized model. Hybrid method.
. [F. Chazel, D. Lannes, F. Marche, 2011] 3 parameters model
. [M. Tissier et al, 2012] Wave breaking issues
. [D. Lannes, F. Marche, 2015] A new class of fully nonlinear and weaklydispersive Green-Naghdi models for efficient 2D simulations
Revoke the time dependency∂tη + ∂x(hu) = 0 ,[1+ αT[hb]
](∂thu + ∂x(hu
2) + α−1α
gh∂xη)+ 1
αgh∂xη
+ h(Q1(u) + gQ2(η)
)+ gQ3
([1+ αT[hb]
]−1(gh∂xη)
)= 0 .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 13 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Model presentation. [D. Lannes, F. Marche, 2015] A new class of fully nonlinear and weaklydispersive Green-Naghdi models for efficient 2D simulations
Revoke the time dependency∂tη + ∂x(hu) = 0 ,[1+ αT[hb]
](∂thu + ∂x(hu
2) + α−1α
gh∂xη)+ 1
αgh∂xη
+ h(Q1(u) + gQ2(η)
)+ gQ3
([1+ αT[hb]
]−1(gh∂xη)
)= 0 .
.T[h]w = −h3
3∂x
2(wh
)− h2∂xh∂x
(wh
), hb = h0 − z ,
.Qi=1,2,3 : non linear, non conservative terms with second order derivatives.
. 2d version : "diagonal" sytem : no coupling between u and v !
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 13 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z) + D(U, z)
. Shallow Water equations :
U =
(hhu
), G (U) =
(hu
12gh2 + hu2
), B(U) =
(0
−gh∂xz
).
. Dispersive terms :
D(V , z) =
(0
Dhu(V , z)
), with
Dhu(V , z) =[1 + αT[hb]
]−1( 1αgh∂xη + h
(Q1(u) + gQ2(η)
)+ gQ3
([1 + αT[hb]
]−1(gh∂xη)
))− 1αgh∂xη .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+D(U, z)
. Shallow Water equations :
U =
(hhu
), G (U) =
(hu
12gh2 + hu2
), B(U) =
(0
−gh∂xz
).
. Dispersive terms :
D(V , z) =
(0
Dhu(V , z)
), with
Dhu(V , z) =[1 + αT[hb]
]−1( 1αgh∂xη + h
(Q1(u) + gQ2(η)
)+ gQ3
([1 + αT[hb]
]−1(gh∂xη)
))− 1αgh∂xη .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+ D(U, z)︸ ︷︷ ︸Dispersive terms
. Shallow Water equations :
U =
(hhu
), G (U) =
(hu
12gh2 + hu2
), B(U) =
(0
−gh∂xz
).
. Dispersive terms :
D(V , z) =
(0
Dhu(V , z)
), with
Dhu(V , z) =[1 + αT[hb]
]−1( 1αgh∂xη + h
(Q1(u) + gQ2(η)
)+ gQ3
([1 + αT[hb]
]−1(gh∂xη)
))− 1αgh∂xη .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+ D(U, z)︸ ︷︷ ︸Dispersive terms
Hyperbolic part : okDispersive part :
Dh(x , t) =
Nd∑l=1
Dl(t)θl(x) , x ∈ Ci .
Well balancing and robustness ,Treatment of the second order derivatives .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+ D(U, z)︸ ︷︷ ︸Dispersive terms
Hyperbolic part : okDispersive part :
Dh(x , t) =
Nd∑l=1
Dl(t)θl(x) , x ∈ Ci .
Well balancing and robustness ,Treatment of the second order derivatives .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+ D(U, z)︸ ︷︷ ︸Dispersive terms
Hyperbolic part : okDispersive part :
Dh(x , t) =
Nd∑l=1
Dl(t)θl(x) , x ∈ Ci .
Well balancing and robustness ,Treatment of the second order derivatives .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Isolation of the hyperbolic part
A convenient formulation
∂tU + ∂xG (U) = B(U, z)︸ ︷︷ ︸Shallow Water
+ D(U, z)︸ ︷︷ ︸Dispersive terms
Hyperbolic part : okDispersive part :
Dh(x , t) =
Nd∑l=1
Dl(t)θl(x) , x ∈ Ci .
Well balancing and robustness ,Treatment of the second order derivatives .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 14 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
LDG formalism
Simplified case : T = ∂x2
Consider the second order ODE :
f − ∂2xu = 0 . (1)
(1) reduces to a coupled system of first order equations.
f + ∂xv = 0 , v + ∂xu = 0 .
Weak formulation∫ x ri
x li
f φh −∫ x r
i
x li
vφ′h + vrφh(x ri )− vlφh(x li ) = 0 ,∫ x ri
x li
vφh −∫ x r
i
x li
uφ′h + urφh(x ri )− ulφh(x li ) = 0 .
LDG schemes :[B. Cockburn, C.-W. Shu, 1998] The Local Discontinuous Galerkin method fortime-dependent convection-diffusion systems
.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 15 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequationsMotivationsThe physicalmodelReformulationof the systemHigh orderderivatives
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
LDG formalism
Simplified case : T = ∂x2
Consider the second order ODE :
f − ∂2xu = 0 . (1)
(1) reduces to a coupled system of first order equations.
f + ∂xv = 0 , v + ∂xu = 0 .
Weak formulation∫ x ri
x li
f φh −∫ x r
i
x li
vφ′h + vrφh(x ri )− vlφh(x li ) = 0 ,∫ x ri
x li
vφh −∫ x r
i
x li
uφ′h + urφh(x ri )− ulφh(x li ) = 0 .
LDG schemes :[B. Cockburn, C.-W. Shu, 1998] The Local Discontinuous Galerkin method fortime-dependent convection-diffusion systems .
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 15 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Outline
1 Shallow Water Equations - Generalities
2 dG discretization
3 Extension to dispersive equations
4 Handling breaking waves (work with G. Richard)
5 Perspectives
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 16 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Switching method : press ENTER and hope ...
Protocol : At each time stepDetection : evaluation of Ik on each cell k .[L. Krivodonova et al, 2004] Shock detection and limiting withdiscontinuous Galerkin methods for hyperbolic conservation laws
Determination of the breaking area.
Switching strategySuppress of the dispersive terms on the targeted area.Application of a limiter to the hyperbolic part (Shallow Water).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 17 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Switching method : press ENTER and hope ...
Protocol : At each time stepDetection : evaluation of Ik on each cell k .[L. Krivodonova et al, 2004] Shock detection and limiting withdiscontinuous Galerkin methods for hyperbolic conservation laws
Determination of the breaking area.
Switching strategySuppress of the dispersive terms on the targeted area.Application of a limiter to the hyperbolic part (Shallow Water).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 17 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Switching method : press ENTER and hope ...
Protocol : At each time stepDetection : evaluation of Ik on each cell k .[L. Krivodonova et al, 2004] Shock detection and limiting withdiscontinuous Galerkin methods for hyperbolic conservation laws
Determination of the breaking area.
Switching strategySuppress of the dispersive terms on the targeted area.Application of a limiter to the hyperbolic part (Shallow Water).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 17 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Switching method : press ENTER and hope ...
Protocol : At each time stepDetection : evaluation of Ik on each cell k .[L. Krivodonova et al, 2004] Shock detection and limiting withdiscontinuous Galerkin methods for hyperbolic conservation laws
Determination of the breaking area.
Switching strategySuppress of the dispersive terms on the targeted area.Application of a limiter to the hyperbolic part (Shallow Water).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 17 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Through a more rigorous approach
. Account for the mechanical energy dissipation through a thirdvariable ϕ.
A new model (G. Richard, 2016)
∂tU + ∂x G (U) = B(U, z) + D(U, z)
with :
U =
hhuhϕ
, G (U) =
hu12gh2 + h3ϕ+ hu2
huϕ
, B(U) =
0−gh∂xz
0
. A simple additional transport equation in the hyperbolic part !