Recent advances on numerical simulation in coastal...
Transcript of Recent advances on numerical simulation in coastal...
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.
Notable advances :. [Greenberg, Leroux, 1996] , [Gosse, Leroux , 1996] scalar case. [Garcia-Navarro, Vázquez-Cendón , 1997] , [Castro, Gonzales, Pares, 2006]Roe schemes, [LeVeque, 1998] wave-propagation algorithm. [Perthame, Simeoni, 2001] , [Perthame, Simeoni, 2003] kinetic schemes. [Gallouët, Hérard, Seguin, 2003] , [Berthon, Marche, 2008] VFRoe schemes. [Audusse et al, 2004] Hydrostatic Reconstruction, [Ricchiuto et al, 2007]RD schemes , [Lukácová-Medvidová, Noelle, Kraft, 2007] FVEG schemes,[Xing, Zhang, Shu, 2010] , [Berthon, Chalons, 2016] ...
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 5 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Reformulation of the SW equations
1D Configuration
z
hu
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 6 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Reformulation of the SW equations
1D Lake at rest configuration
z
h
η = cte
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 6 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Reformulation of the SW equations
1D Lake at rest configuration
z
h
η = cte
First works :. [Zhou, Causon, Mingham, 2001] Surface Gradient Method. [Rogers, Fujihara, Borthwick, 2001] , [Russo, 2005] , [Xing, Shu, 2005]
. [Liang, Borthwick, 2009] , [Liang, Marche, 2009] “Pre-Balanced”formulation.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 6 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Reformulation of the SW equations
1D Lake at rest configuration
z
h
η = cte
First works :. [Zhou, Causon, Mingham, 2001] Surface Gradient Method. [Rogers, Fujihara, Borthwick, 2001] , [Russo, 2005] , [Xing, Shu, 2005]. [Liang, Borthwick, 2009] , [Liang, Marche, 2009] “Pre-Balanced”formulation.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 6 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -GeneralitiesIntroductionNumericalstabilitycriteriaReformulationof the SWequations
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Reformulation of the SW equations
1D Lake at rest configuration
z
h
η = cte
Pre balanced formulation
∂tV +∇.H(V , z) = S(V , z) .
V =
η
hu
hv
, H(V , z) =
hu hv12g(η
2 − 2ηz) + hu2 huv
huv 12g(η
2 − 2ηz) + hv2
.
Topography source term : S(V , z) =
0−gη∂xz−gη∂yz
.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 6 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Outline
1 Shallow Water Equations - Generalities
2 dG discretizationdG method : generalitiesNumerical fluxesPreservation of the water depth positivity
3 Extension to dispersive equations
4 Handling breaking waves (work with G. Richard)
5 Perspectives
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 7 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Local weak formulation (1)
∂tV +∇.H(V , z) = S(V , z)
∫T
∂tVφh(x)dx−∫T
H(V , z).∇φh(x)dx+∫∂T
H(V , z).~nφh(s)ds =
∫T
S(V , z)φh(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Local weak formulation (2)
∂tVφh(x) +∇.H(V , z)φh(x) = S(V , z)φh(x)
∫T
∂tVφh(x)dx−∫T
H(V , z).∇φh(x)dx+∫∂T
H(V , z).~nφh(s)ds =
∫T
S(V , z)φh(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Local weak formulation (3)∫T
∂tVφh(x)dx +
∫T
∇.H(V , z)φh(x)dx =
∫T
S(V , z)φh(x)dx
∫T
∂tVφh(x)dx−∫T
H(V , z).∇φh(x)dx+∫∂T
H(V , z).~nφh(s)ds =
∫T
S(V , z)φh(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Local weak formulation (4)∫T
∂tVφh(x)dx +∫T∇.H(V , z)φh(x)dx =
∫T
S(V , z)φh(x)dx
∫T
∂tVφh(x)dx−∫T
H(V , z).∇φh(x)dx+∫∂T
H(V , z).~nφh(s)ds =
∫T
S(V , z)φh(x)dx
@@R
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Through a semi-discrete formulation (1)
•V → Vh
•φh → θj
∫T
∂t
( Nd∑l=1
Vl(t)θl(x))φh(x)dx−
∫T
H(Vh, zh).∇φh(x)dx+∫∂T
H(Vh, zh).~nφh(s)ds =
∫T
S(Vh, zh)φh(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Through a semi-discrete formulation (2)
•V → Vh
•φh → θj∫T
∂t
( Nd∑l=1
Vl(t)θl(x))θj(x)dx−
∫T
H(Vh, zh).∇θj(x)dx+∫∂T
H(Vh, zh).~nθj(s)ds =
∫T
S(Vh, zh)θj(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
dG method : general background
Pd(T ) := 2 variables polynomials onTof degree at most d .
Vh := v ∈ L2(Ω) | ∀T ∈ Th , v|T ∈ Pd(T ).
Approximate solution : Vh(x, t) =
Nd∑l=1
Vl(t)θl(x) , ∀x ∈ T .
Through a semi-discrete formulation•V → Vh
•φh → θj∫T
∂t
( Nd∑l=1
Vl(t)θl(x))θj(x)dx−
∫T
H(Vh, zh).∇θj(x)dx+
3∑k=1
∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds =
∫T
S(Vh, zh)θj(x)dx
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 8 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
DG
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF DG
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF DG
→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Numerical fluxes
Contributions on the edges∫Γij(k)
H(Vh, zh).~nij(k)θj(s)ds .
H(Vh, zh).~nij(k) ≈ Hij(k) = H(V−k , V+k , zk , zk , ~nij(k)) .
VF DG→ Preservation of the motionless steady states.
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 9 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Preservation of the water depth positivity
dG schemes and maximum principle[X. Zhang, C.-W. Shu, 2010] Maximum-principle-satisfying high orderschemes - 1d and 2d structured meshes
[Y. Xing, X. Zhang, C.-W. Shu, 2010] Application to 1d SW
[Y. Xing, X. Zhang, 2013] Extension to triangular meshes
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 10 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretizationdG method :generalitiesNumericalfluxesPreservationof the waterdepthpositivity
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Preservation of the water depth positivity
dG schemes and maximum principle[X. Zhang, C.-W. Shu, 2010] Maximum-principle-satisfying high orderschemes - 1d and 2d structured meshes
[Y. Xing, X. Zhang, C.-W. Shu, 2010] Application to 1d SW
[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 ).
[Kazolea, Delis, Synolakis] (FV, 2014),[Filippini, Kazolea, Ricchiuto] (Hybrid FV/FE, 2016)
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 12 / 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 equationsNumerical issues. Non conservative terms, high order derivatives,wave-breaking, non linearities.
. Maintain the stability of the method (positivity, well balancing),even on unstructured environments.
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 ).
[Kazolea, Delis, Synolakis] (FV, 2014),[Filippini, Kazolea, Ricchiuto] (Hybrid FV/FE, 2016)
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 12 / 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 equationsNumerical issues. Non conservative terms, high order derivatives,wave-breaking, non linearities.
. Maintain the stability of the method (positivity, well balancing),even on unstructured environments.
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 ).
[Kazolea, Delis, Synolakis] (FV, 2014),[Filippini, Kazolea, Ricchiuto] (Hybrid FV/FE, 2016)
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 12 / 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
. [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 !
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 18 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Convergence analysis and model comparison
Profiles
Convergence ratesNe
N 20 40 80 160 320 order1 2.5e-1 4.2e-2 1.0e-3 2.8e-3 9.6e-4 1.92 7.5e-2 7.5e-3 6.2e-4 6.5e-5 7.7e-6 3.23 4.5e-3 3.0e-4 1.7e-5 9.4e-7 5.7e-8 4.04 7.0e-4 1.6e-5 4.6e-7 1.4e-8 4.4e-10 5.15 6.1e-5 7.6e-7 1.0e-8 1.6e-10 3.1e-12 6.1
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 19 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Convergence analysis and model comparison
Profiles
Convergence ratesNe
N 20 40 80 160 320 order1 2.5e-1 4.2e-2 1.0e-3 2.8e-3 9.6e-4 1.92 7.5e-2 7.5e-3 6.2e-4 6.5e-5 7.7e-6 3.23 4.5e-3 3.0e-4 1.7e-5 9.4e-7 5.7e-8 4.04 7.0e-4 1.6e-5 4.6e-7 1.4e-8 4.4e-10 5.15 6.1e-5 7.6e-7 1.0e-8 1.6e-10 3.1e-12 6.1
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 19 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
CPU time
Evolution of the ratio τ = ρo/ρc .ρ : mean iteration time (based on 1000 iterations).
Ne
N 1000 2000 3000 4000 5000 60001 3.23 3.21 3.04 2.96 2.94 2.902 4.09 4.27 4.14 4.01 3.90 3.843 5.32 5.11 5.03 4.97 4.91 4.874 6.01 5.77 5.67 5.63 5.51 5.555 6.66 6.38 6.32 6.30 6.26 6.166 7.15 6.99 7.05 6.97 6.86 6.54
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 20 / 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 21 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Perspectives
Boundary conditions→ Modelling, Numerical methods.→ Theoretical investigations.
Handling breaking waves→ Coupling SW/GN, smoothness criteria.→ Works of S. Gavrilyuk and collaboration with G. Richard (inprogress.)
Numerical treatment of the non-hydrostatic terms→ Weak formulations.→ Numerical exploration (linear solvers, matrix storage,re-numbering).→ Alternative approaches (quadrature rules, Finite Volume methods,FEM).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 22 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Perspectives
Boundary conditions→ Modelling, Numerical methods.→ Theoretical investigations.
Handling breaking waves→ Coupling SW/GN, smoothness criteria.→ Works of S. Gavrilyuk and collaboration with G. Richard (inprogress.)
Numerical treatment of the non-hydrostatic terms→ Weak formulations.→ Numerical exploration (linear solvers, matrix storage,re-numbering).→ Alternative approaches (quadrature rules, Finite Volume methods,FEM).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 22 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Perspectives
Boundary conditions→ Modelling, Numerical methods.→ Theoretical investigations.
Handling breaking waves→ Coupling SW/GN, smoothness criteria.→ Works of S. Gavrilyuk and collaboration with G. Richard (inprogress.)
Numerical treatment of the non-hydrostatic terms→ Weak formulations.→ Numerical exploration (linear solvers, matrix storage,re-numbering).→ Alternative approaches (quadrature rules, Finite Volume methods,FEM).
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 22 / 23
SHARK-FV
Arnaud Duran
Shallow WaterEquations -Generalities
dGdiscretization
Extension todispersiveequations
Handlingbreaking waves(work with G.Richard)
Numericalvalidations
Perspectives
Thanks !
Arnaud Duran (ICJ) SHARK-FV 19/05/2017 23 / 23