SHARK-FV Arnaud Duran
Recent advances on numerical simulation in coastal oceanography
Shallow Water Equations Generalities dG discretization Extension to dispersive equations
Arnaud Duran Institut Camille Jordan - Université Claude Bernard Lyon 1
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
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
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities
Shallow Water Equations Generalities dG discretization
2
dG discretization
3
Extension to dispersive equations
Handling breaking waves (work with G. Richard)
4
Handling breaking waves (work with G. Richard)
Numerical validations
5
Perspectives
Extension to dispersive equations
Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
2 / 23
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities Introduction Numerical stability criteria Reformulation of the SW equations
Shallow Water Equations Generalities Introduction Numerical stability criteria Reformulation of the SW equations
2
dG discretization
3
Extension to dispersive equations
Extension to dispersive equations
4
Handling breaking waves (work with G. Richard)
Handling breaking waves (work with G. Richard)
5
Perspectives
Numerical validations
dG discretization
Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
3 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard)
Coastal hydrodynamic
Arnaud Duran (ICJ)
Shallow Water Equations Generalities
SHARK-FV
Numerical validations Perspectives
19/05/2017
4 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard)
Coastal hydrodynamic
Arnaud Duran (ICJ)
Shallow Water Equations Generalities
SHARK-FV
Numerical validations Perspectives
19/05/2017
4 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard)
Coastal hydrodynamic
Arnaud Duran (ICJ)
Shallow Water Equations Generalities
SHARK-FV
Numerical validations Perspectives
19/05/2017
4 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard)
Coastal hydrodynamic
Arnaud Duran (ICJ)
Shallow Water Equations Generalities
SHARK-FV
Numerical validations Perspectives
19/05/2017
4 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
Shallow Water Equations Generalities
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard) Numerical validations
Tsunamis
Arnaud Duran (ICJ)
SHARK-FV
Perspectives
19/05/2017
4 / 23
Shallow Water Equations SHARK-FV
2D Formulation
Arnaud Duran
∂t U + ∇.G (U) = B(U, z) .
hu
h 1 2 2 U = hu , G (U) = 2 gh + hu hv huv
hv
0 , B(U, z) = −gh∂x z 1 2 −gh∂y z gh + hv 2 2 huv
Introduction Numerical stability criteria Reformulation of the SW equations dG discretization Extension to dispersive equations
Application field : some examples
Handling breaking waves (work with G. Richard)
Rivers, dam breaks
Arnaud Duran (ICJ)
Shallow Water Equations Generalities
SHARK-FV
Numerical validations Perspectives
19/05/2017
4 / 23
Numerical issues SHARK-FV
Stability criteria
Arnaud Duran
Preservation of steady states : → (C-property) [Bermudez & Vázquez, 1994]
Shallow Water Equations Generalities Introduction Numerical stability criteria Reformulation of the SW equations
h + z = cte , u = 0 . Robustness : preservation of the water depth positivity.
dG discretization
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
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
5 / 23
Reformulation of the SW equations SHARK-FV Arnaud Duran Shallow Water Equations Generalities Introduction Numerical stability criteria Reformulation of the SW equations
1D Configuration h
u
dG discretization Extension to dispersive equations
z
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
6 / 23
Reformulation of the SW equations SHARK-FV Arnaud Duran Shallow Water Equations Generalities Introduction Numerical stability criteria Reformulation of the SW equations
1D Lake at rest configuration
dG discretization
h z
η = cte
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
6 / 23
Reformulation of the SW equations SHARK-FV Arnaud Duran Shallow Water Equations Generalities
1D Lake at rest configuration
Introduction Numerical stability criteria Reformulation of the SW equations
h z
dG discretization
η = cte
First works : . [Zhou, Causon, Mingham, 2001] Surface Gradient Method . [Rogers, Fujihara, Borthwick, 2001] , [Russo, 2005] , [Xing, Shu, 2005]
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
6 / 23
Reformulation of the SW equations SHARK-FV Arnaud Duran Shallow Water Equations Generalities
1D Lake at rest configuration
Introduction Numerical stability criteria Reformulation of the SW equations
h z
dG discretization
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
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
6 / 23
Reformulation of the SW equations SHARK-FV
1D Lake at rest configuration
Arnaud Duran Shallow Water Equations Generalities
h z
Introduction Numerical stability criteria Reformulation of the SW equations
η = cte
dG discretization
Pre balanced formulation
Extension to dispersive equations
∂t V + ∇.H(V , z) = S(V , z) .
η hu hv V = hu , H(V , z) = 12 g (η 2 − 2ηz) + hu 2 huv . 2 2 1 huv hv g (η − 2ηz) + hv 2 0 Topography source term : S(V , z) = −g η∂x z . −g η∂y z Arnaud Duran (ICJ)
SHARK-FV
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
6 / 23
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities
2
dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity
3
Extension to dispersive equations
Extension to dispersive equations
4
Handling breaking waves (work with G. Richard)
Handling breaking waves (work with G. Richard)
5
Perspectives
Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
7 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on
Arnaud Duran
T of degree at most d} . Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
l=1
Local weak formulation (1) ∂t V + ∇.H(V , z) = S(V , z)
∀x ∈ T .
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on
Arnaud Duran
T of degree at most d} . Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
l=1
Local weak formulation (2) ∂t V φh (x) + ∇.H(V , z)φh (x) = S(V , z)φh (x)
∀x ∈ T .
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on
Arnaud Duran
T of degree at most d} . Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
∀x ∈ T .
l=1
Local weak formulation (3) Z
Z ∇.H(V , z)φh (x)dx =
∂t V φh (x)dx + T
Z
T
S(V , z)φh (x)dx T
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on
Arnaud Duran
T of degree at most d} . Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
dG method : generalities Numerical fluxes Preservation of the water depth positivity
∀x ∈ T .
l=1
Local weak formulation (4) Z ∂t V φh (x)dx + T
Z T
Z
R T
∇.H(V , z)φh (x)dx =
S(V , z)φh (x)dx T
@ R @ ∂t V φh (x)dx − H(V , z).∇φh (x)dx + T Z Z H(V , z).~ nφh (s)ds = S(V , z)φh (x)dx Z
∂T
Arnaud Duran (ICJ)
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
T
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on T of degree at most d} .
Arnaud Duran Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
∀x ∈ T .
l=1
Through a semi-discrete formulation (1)
Extension to dispersive equations
• V → Vh Z ∂t T
dG method : generalities Numerical fluxes Preservation of the water depth positivity
Nd X
Z Vl (t)θl (x) φh (x)dx − H(Vh , zh ).∇φh (x)dx + T
l=1
Z
Z H(Vh , zh ).~ nφh (s)ds =
∂T
Arnaud Duran (ICJ)
S(Vh , zh )φh (x)dx
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
T
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on T of degree at most d} .
Arnaud Duran Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
∀x ∈ T .
l=1
Through a semi-discrete formulation (2) • V → Vh • φh → θ j Z Z Nd X Vl (t)θl (x) θj (x)dx − H(Vh , zh ).∇θj (x)dx + ∂t T
T
l=1
Z
Z H(Vh , zh ).~ nθj (s)ds =
∂T
Arnaud Duran (ICJ)
S(Vh , zh )θj (x)dx
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
T
SHARK-FV
19/05/2017
8 / 23
dG method : general background SHARK-FV
Pd (T ) := {2 variables polynomials on
Arnaud Duran
T of degree at most d} .
Shallow Water Equations Generalities
Vh := {v ∈ L2 (Ω) | ∀T ∈ Th , v|T ∈ Pd (T )}. Approximate solution : Vh (x, t) =
Nd X
dG discretization
Vl (t)θl (x) ,
∀x ∈ T .
l=1
Through a semi-discrete formulation • V → Vh • φh → θ j Z Z Nd X ∂t Vl (t)θl (x) θj (x)dx − H(Vh , zh ).∇θj (x)dx + T
k=1
T
l=1
3 Z X
H(Vh , zh ).~ nij(k) θj (s)ds =
Arnaud Duran (ICJ)
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations
Z
Γij(k)
dG method : generalities Numerical fluxes Preservation of the water depth positivity
Perspectives
S(Vh , zh )θj (x)dx T
SHARK-FV
19/05/2017
8 / 23
Numerical fluxes SHARK-FV Arnaud Duran Shallow Water Equations Generalities
Contributions on the edges
dG discretization
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes SHARK-FV Arnaud Duran Shallow Water Equations Generalities
Contributions on the edges
dG discretization
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes SHARK-FV
Contributions on the edges
Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes SHARK-FV
Contributions on the edges
Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations
VF
Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes SHARK-FV
Contributions on the edges
Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard)
DG
Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes Contributions on the edges
SHARK-FV Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
VF Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes Contributions on the edges
SHARK-FV Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
VF Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes Contributions on the edges
SHARK-FV Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
VF
DG Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes Contributions on the edges
SHARK-FV Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
VF
DG Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Numerical fluxes Contributions on the edges
SHARK-FV Arnaud Duran
Z H(Vh , zh ).~ nij(k) θj (s)ds . Γij(k)
H(Vh , zh ).~ nij(k) ≈ Hij(k) = H(V˘k− , V˘k+ , z˘k , z˘k , n~ij(k) ) .
Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
VF DG → Preservation of the motionless steady states. Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
9 / 23
Preservation of the water depth positivity SHARK-FV Arnaud Duran Shallow Water Equations Generalities dG discretization
dG schemes and maximum principle [X. Zhang, C.-W. Shu, 2010] Maximum-principle-satisfying high order schemes - 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
dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
10 / 23
Preservation of the water depth positivity dG schemes and maximum principle [X. Zhang, C.-W. Shu, 2010] Maximum-principle-satisfying high order schemes - 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 method relies on a special quadrature rule.
SHARK-FV Arnaud Duran Shallow Water Equations Generalities dG discretization dG method : generalities Numerical fluxes Preservation of the water depth positivity Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations
Figure: Nodes locations for the special quadrature - P2 and P3
Perspectives
reduces to the study of a convex combination of first order Finite Volume schemes. Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
10 / 23
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities
Shallow Water Equations Generalities
2
dG discretization
dG discretization
3
Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives
Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives
4
Handling breaking waves (work with G. Richard)
5
Perspectives
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
11 / 23
Interest of dispersive equations SHARK-FV
Objective
Arnaud Duran
Extend the range of applicability of the computations at coast. . Describe the non-linearities before the breaking point. . Dispersive equations : O(µ2 )-accurate Shallow Water equations : O(µ)-accurate .
Shallowness parameter : µ =
h02 . λ20
Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Dispersive equations
Arnaud Duran (ICJ)
Shallow Water
SHARK-FV
19/05/2017
12 / 23
Interest of dispersive equations Numerical issues
SHARK-FV
. Non conservative terms, high order derivatives, wave-breaking, non linearities. . Maintain the stability of the method (positivity, well balancing), even on unstructured environments.
Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives
Dispersive equations
Shallow Water
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
12 / 23
Interest of dispersive equations Numerical issues
SHARK-FV
. Non conservative terms, high order derivatives, wave-breaking, non linearities. . Maintain the stability of the method (positivity, well balancing), even on unstructured environments.
Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives
Dispersive equations
Handling breaking waves (work with G. Richard)
Shallow Water
State of the art
Numerical validations
. 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
Perspectives
19/05/2017
12 / 23
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 weakly dispersive Green-Naghdi models for efficient 2D simulations
SHARK-FV Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard)
Revoke the time dependency
Numerical validations
∂t η + ∂x (hu)= 0 , 1 + αT[hb ] ∂t hu + ∂x (hu 2 ) + α−1 gh∂x η + α1 gh∂x η α −1 + h Q1 (u) + g Q2 (η) + g Q3 1 + αT[hb ] (gh∂x η) = 0 . Arnaud Duran (ICJ)
SHARK-FV
Perspectives
19/05/2017
13 / 23
Model presentation . [D. Lannes, F. Marche, 2015] A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations
SHARK-FV Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives
Revoke the time dependency ∂t η + ∂x (hu)= 0 , 1 + αT[hb ] ∂t hu + ∂x (hu 2 ) + α−1 gh∂x η + α1 gh∂x η α −1 + h Q1 (u) + g Q2 (η) + g Q3 1 + αT[hb ] (gh∂x η) = 0 . w h3 2 w ∂x − h2 ∂x h∂x , hb = h0 − z , 3 h h : non linear, non conservative terms with second order derivatives.
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
. T[h]w = − . Qi=1,2,3
. 2d version : "diagonal" sytem : no coupling between u and v ! Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
13 / 23
Isolation of the hyperbolic part SHARK-FV
A convenient formulation
Arnaud Duran
∂t U + ∂x G (U) = B(U, z) + D(U, z)
Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV
A convenient formulation
Arnaud Duran
∂t U + ∂x G (U) = B(U, z) +D(U, z) {z } |
Shallow Water Equations Generalities
Shallow Water
dG discretization
. Shallow Water equations : ! hu h U= , G (U) = 1 2 hu gh + hu 2 2
Extension to dispersive equations
,
B(U) =
0 −gh∂x z
.
Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV
A convenient formulation
Arnaud Duran
∂t U + ∂x G (U) = B(U, z) + | {z } Shallow Water
D(U, z) | {z }
Shallow Water Equations Generalities
Dispersive terms
dG discretization
. Shallow Water equations : U=
h hu
,
G (U) =
hu 1 2 gh + hu 2 2
!
,
B(U) =
0 −gh∂x z
. Dispersive terms : 0 D(V , z) = , with Dhu (V , z) −1 1 Dhu (V , z) = 1 + αT[hb ] gh∂x η + h Q1 (u) + g Q2 (η) α −1 1 + g Q3 1 + αT[hb ] (gh∂x η) − gh∂x η . α Arnaud Duran (ICJ)
SHARK-FV
Extension to dispersive equations
.
Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV Arnaud Duran
A convenient formulation ∂t U + ∂x G (U) = B(U, z) + | {z } Shallow Water
D(U, z) | {z }
Dispersive terms
Shallow Water Equations Generalities dG discretization Extension to dispersive equations
Hyperbolic part : ok
Motivations The physical model Reformulation of the system High order derivatives
Dispersive part : Dh (x, t) =
Nd X
Dl (t)θl (x) , x ∈ Ci .
l=1
Well balancing and robustness , Treatment of the second order derivatives .
Arnaud Duran (ICJ)
SHARK-FV
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV Arnaud Duran
A convenient formulation ∂t U + ∂x G (U) = B(U, z) + | {z } Shallow Water
D(U, z) | {z }
Dispersive terms
Shallow Water Equations Generalities dG discretization Extension to dispersive equations
Hyperbolic part : ok
Motivations The physical model Reformulation of the system High order derivatives
Dispersive part : Dh (x, t) =
Nd X
Dl (t)θl (x) , x ∈ Ci .
l=1
Well balancing and robustness , Treatment of the second order derivatives .
Arnaud Duran (ICJ)
SHARK-FV
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV Arnaud Duran
A convenient formulation ∂t U + ∂x G (U) = B(U, z) + | {z } Shallow Water
D(U, z) | {z }
Dispersive terms
Shallow Water Equations Generalities dG discretization Extension to dispersive equations
Hyperbolic part : ok
Motivations The physical model Reformulation of the system High order derivatives
Dispersive part : Dh (x, t) =
Nd X
Dl (t)θl (x) , x ∈ Ci .
l=1
Well balancing and robustness , Treatment of the second order derivatives .
Arnaud Duran (ICJ)
SHARK-FV
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
14 / 23
Isolation of the hyperbolic part SHARK-FV Arnaud Duran
A convenient formulation ∂t U + ∂x G (U) = B(U, z) + | {z } Shallow Water
D(U, z) | {z }
Dispersive terms
Shallow Water Equations Generalities dG discretization Extension to dispersive equations
Hyperbolic part : ok
Motivations The physical model Reformulation of the system High order derivatives
Dispersive part : Dh (x, t) =
Nd X
Dl (t)θl (x) , x ∈ Ci .
l=1
Well balancing and robustness , Treatment of the second order derivatives .
Arnaud Duran (ICJ)
SHARK-FV
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
14 / 23
LDG formalism SHARK-FV
Simplified case : T = ∂x 2
Arnaud Duran
Consider the second order ODE : f − ∂x2 u = 0 . (1) reduces to a coupled system of first order equations. f + ∂x v = 0 ,
v + ∂x u = 0 .
Weak formulation xir
Z
xir
Z f φh −
xil
Z
xil
xir
Z
xir
v φh − xil
xil
v φ0h + vbr φh (xir ) − vbl φh (xil ) = 0 , uφ0h + ubr φh (xir ) − ubl φh (xil ) = 0 .
(1)
Shallow Water Equations Generalities dG discretization Extension to dispersive equations Motivations The physical model Reformulation of the system High order derivatives Handling breaking waves (work with G. Richard) Numerical validations Perspectives
. Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
15 / 23
LDG formalism SHARK-FV
Simplified case : T = ∂x 2
Arnaud Duran
Consider the second order ODE : f − ∂x2 u = 0 .
(1)
dG discretization
(1) reduces to a coupled system of first order equations. f + ∂x v = 0 ,
Extension to dispersive equations
v + ∂x u = 0 .
Motivations The physical model Reformulation of the system High order derivatives
Weak formulation xir
Z
xir
Z f φh −
xil
Z
xil
xir
Z
xir
v φh − xil
xil
Shallow Water Equations Generalities
v φ0h + vbr φh (xir ) − vbl φh (xil ) = 0 ,
Handling breaking waves (work with G. Richard)
uφ0h + ubr φh (xir ) − ubl φh (xil ) = 0 .
Numerical validations Perspectives
LDG schemes : [B. Cockburn, C.-W. Shu, 1998] The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems . Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
15 / 23
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities
Shallow Water Equations Generalities dG discretization
2
dG discretization
3
Extension to dispersive equations
4
Handling breaking waves (work with G. Richard)
5
Perspectives
Arnaud Duran (ICJ)
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
SHARK-FV
19/05/2017
16 / 23
Switching method : press ENTER and hope ... SHARK-FV Arnaud Duran
Protocol : At each time step
Shallow Water Equations Generalities
Detection : evaluation of Ik on each cell k. [L. Krivodonova et al, 2004] Shock detection and limiting with
dG discretization
discontinuous Galerkin methods for hyperbolic conservation laws
Extension to dispersive equations
Determination of the breaking area.
Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Switching strategy Suppress 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
Switching method : press ENTER and hope ... SHARK-FV Arnaud Duran
Protocol : At each time step Detection : evaluation of Ik on each cell k.
Shallow Water Equations Generalities
[L. Krivodonova et al, 2004] Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
dG discretization
Determination of the breaking area.
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Switching strategy Suppress 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
Switching method : press ENTER and hope ... SHARK-FV Arnaud Duran
Protocol : At each time step Detection : evaluation of Ik on each cell k.
Shallow Water Equations Generalities
[L. Krivodonova et al, 2004] Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
dG discretization
Determination of the breaking area.
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Switching strategy Suppress 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
Switching method : press ENTER and hope ... SHARK-FV Arnaud Duran
Protocol : At each time step Detection : evaluation of Ik on each cell k.
Shallow Water Equations Generalities
[L. Krivodonova et al, 2004] Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
dG discretization
Determination of the breaking area.
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Switching strategy Suppress 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
Through a more rigorous approach SHARK-FV Arnaud Duran
. Account for the mechanical energy dissipation through a third variable ϕ.
Shallow Water Equations Generalities dG discretization
A new model (G. Richard, 2016)
Extension to dispersive equations
˜ = B(U, ˜ z) + D(U, ˜ z) ∂t U˜ + ∂x G˜(U)
Handling breaking waves (work with G. Richard)
with : hu h 0 1 ˜ = ˜ = −gh∂x z U˜ = hu , G˜(U) gh2 + h3 ϕ + hu 2 , B(U) 2 hϕ 0 huϕ
Numerical validations Perspectives
. A simple additional transport equation in the hyperbolic part !
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
18 / 23
Convergence analysis and model comparison SHARK-FV
Profiles
Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Handling breaking waves (work with G. Richard)
Convergence rates N 1 2 3 4 5
20 2.5e-1 7.5e-2 4.5e-3 7.0e-4 6.1e-5
Arnaud Duran (ICJ)
40 4.2e-2 7.5e-3 3.0e-4 1.6e-5 7.6e-7
Ne 80 1.0e-3 6.2e-4 1.7e-5 4.6e-7 1.0e-8
Numerical validations Perspectives
160 2.8e-3 6.5e-5 9.4e-7 1.4e-8 1.6e-10 SHARK-FV
320 9.6e-4 7.7e-6 5.7e-8 4.4e-10 3.1e-12
order 1.9 3.2 4.0 5.1 6.1 19/05/2017
19 / 23
Convergence analysis and model comparison SHARK-FV
Profiles
Arnaud Duran Shallow Water Equations Generalities dG discretization Extension to dispersive equations Handling breaking waves (work with G. Richard)
Convergence rates N 1 2 3 4 5
20 2.5e-1 7.5e-2 4.5e-3 7.0e-4 6.1e-5
Arnaud Duran (ICJ)
40 4.2e-2 7.5e-3 3.0e-4 1.6e-5 7.6e-7
Ne 80 1.0e-3 6.2e-4 1.7e-5 4.6e-7 1.0e-8
Numerical validations Perspectives
160 2.8e-3 6.5e-5 9.4e-7 1.4e-8 1.6e-10 SHARK-FV
320 9.6e-4 7.7e-6 5.7e-8 4.4e-10 3.1e-12
order 1.9 3.2 4.0 5.1 6.1 19/05/2017
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CPU time SHARK-FV Arnaud Duran Shallow Water Equations Generalities
Evolution of the ratio τ = ρo /ρc . ρ : mean iteration time (based on 1000 iterations).
dG discretization
Ne N 1 2 3 4 5 6
1000 3.23 4.09 5.32 6.01 6.66 7.15
Arnaud Duran (ICJ)
2000 3.21 4.27 5.11 5.77 6.38 6.99
3000 3.04 4.14 5.03 5.67 6.32 7.05
4000 2.96 4.01 4.97 5.63 6.30 6.97
SHARK-FV
5000 2.94 3.90 4.91 5.51 6.26 6.86
6000 2.90 3.84 4.87 5.55 6.16 6.54
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
19/05/2017
20 / 23
Outline SHARK-FV Arnaud Duran
1
Shallow Water Equations - Generalities
Shallow Water Equations Generalities dG discretization
2
dG discretization
3
Extension to dispersive equations
4
Handling breaking waves (work with G. Richard)
5
Perspectives
Arnaud Duran (ICJ)
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
SHARK-FV
19/05/2017
21 / 23
Perspectives SHARK-FV
Boundary conditions
Arnaud Duran
→ Modelling, Numerical methods. → Theoretical investigations.
Shallow Water Equations Generalities dG discretization Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
22 / 23
Perspectives SHARK-FV
Boundary conditions
Arnaud Duran
→ Modelling, Numerical methods. → Theoretical investigations.
Shallow Water Equations Generalities dG discretization
Handling breaking waves → Coupling SW/GN, smoothness criteria. → Works of S. Gavrilyuk and collaboration with G. Richard (in progress.)
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
22 / 23
Perspectives SHARK-FV
Boundary conditions
Arnaud Duran
→ Modelling, Numerical methods. → Theoretical investigations.
Shallow Water Equations Generalities dG discretization
Handling breaking waves
Extension to dispersive equations
→ Coupling SW/GN, smoothness criteria. → Works of S. Gavrilyuk and collaboration with G. Richard (in progress.)
Handling breaking waves (work with G. Richard) Numerical validations
Numerical treatment of the non-hydrostatic terms
Perspectives
→ 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 Water Equations Generalities dG discretization
Thanks !
Extension to dispersive equations Handling breaking waves (work with G. Richard) Numerical validations Perspectives
Arnaud Duran (ICJ)
SHARK-FV
19/05/2017
23 / 23
