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Non-hydrostatic shallow water model and Gradient
Discretization Method
V. Dubos, C. Guichard, Y. Penel and J. Sainte-Marie
LJLL, Sorbonne Université & ANGE, Inria
October 16, 2018
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 1 / 19
Outline
1 Dispersive eects modellingOverview of free-surface owsA depth-averaged Euler modelNumerical analysis
2 Gradient Discretization MethodProblematic and objectiveExample: linear stationary diusion problemThe conforming P1 Finite Elements case
3 Application of GDM to the elliptic problemGradient scheme for the elliptic problemGDM for mixed BCs (homo. Dirichlet)
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 2 / 19
Free-surface ow modelling
Free-surfaceincompressibleEuler equations
Shallow water
equations
Hydrostatic pressure
Homogeneous velocity
Shallow water assumption
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 3 / 19
Free-surface ow modelling
Free-surfaceincompressibleEuler equations
Shallow water
equations
hhhhhhhhhHydrostatic pressure
Homogeneous velocity
Shallow water assumption
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 3 / 19
Emphasis of non-hydrostatic eects
M.-W. Dingemans, Wave propagation over uneven bottoms (Adv. Ser.Ocean Eng., 1997)
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 4 / 19
Emphasis of non-hydrostatic eects
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 4 / 19
A depth-averaged Euler model
PhD of Nora Aïssiouene(LJLL, 2016) :
water height H
velocity u = (v ,w)
(non-hydrostatic) pressure pnh
α = 2 (α = 32:
Serre-Green-Naghdi)
N. Aïssiouene, M.-O. Bristeau, E. Godlewski, J. Sainte-Marie, A combined nite
volume nite element scheme for a dispersive shallow water system (Netw.Heterog. Media 11(1), 2016)
N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés, J.Sainte-Marie, A two-dimensional method for a dispersive shallow water model
(submitted)
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 5 / 19
A depth-averaged Euler model
PhD of Nora Aïssiouene(LJLL, 2016) :
water height H
velocity u = (v ,w)
(non-hydrostatic) pressure pnh
α = 2 (α = 32:
Serre-Green-Naghdi)∂H
∂t+ ∇ · (Hu) = 0
∂(Hu)
∂t+ ∇ · (Hu⊗ u) + ∇
(gH2
2
)+ ∇αsw pnh + gH∇zb = 0
divαsw u = 0
Notations
u =
(v
w
)∇αsw p =
(H∇xp + p∇x(H + 2zb)
−αp
)∇ =
(∇x
0
)div
αsw u = ∇x · (H v)− v · ∇x(H + 2zb) + αw
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 5 / 19
A depth-averaged Euler model
PhD of Nora Aïssiouene(LJLL, 2016) :
water height H
velocity u = (v ,w)
(non-hydrostatic) pressure pnh
α = 2 (α = 32:
Serre-Green-Naghdi)
∂H
∂t+ ∇ · (Hu) = 0
∂(Hu)
∂t+ ∇ · (Hu⊗ u) + ∇
(gH2
2
)+ ∇αsw pnh + gH∇zb = 0
divαsw u = 0
Numerical strategy: Time splitting (projection/correction) Hyperbolic solver /Dispersive solver
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 5 / 19
A depth-averaged Euler model
PhD of Nora Aïssiouene(LJLL, 2016) :
water height H
velocity u = (v ,w)
(non-hydrostatic) pressure pnh
α = 2 (α = 32:
Serre-Green-Naghdi)
∂H
∂t+ ∇ · (Hu) = 0
∂(Hu)
∂t+ ∇ · (Hu⊗ u) + ∇
(gH2
2
)+ ∇αsw pnh + gH∇zb = 0
divαsw u = 0
Numerical strategy: Time splitting (projection/correction) Hyperbolic solver /Dispersive solver
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 5 / 19
Correction step
Focus on the elliptic part - Ω ⊂ Rd (d = 1 or d = 2)
Find pnh : Ω→ R and u : Ω→ Rd+1 s.t.Hu +∇αsw pnh = g on Ω
divαsw u = f on Ω
Hu · ns = φ on Γn
pnh = 0 on Γd
with
ζ := H + 2zb ∇αsw pnh = (H∇pnh + pnh∇ζ,−αpnh)
∂Ω = Γ = Γd ∪ Γn divαsw u = div(Hv)− v · ∇ζ + αw
ns = (nΓn , 0)
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 6 / 19
Correction step
Focus on the elliptic part - Ω ⊂ Rd (d = 1 or d = 2)
Find pnh : Ω→ R and u : Ω→ Rd+1 s.t.Hu +∇αsw pnh = g on Ω
divαsw u = f on Ω
Hu · ns = φ on Γn
pnh = 0 on Γd
with
ζ := H + 2zb ∇αsw pnh = (H∇pnh + pnh∇ζ,−αpnh)
∂Ω = Γ = Γd ∪ Γn divαsw u = div(Hv)− v · ∇ζ + αw
ns = (nΓn , 0)
Stokes-type formula :∫Ω
∇αsw pnh · u = −∫
Ω
pnh divαsw u +
∫∂Ω
γpnhHu · ns
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 6 / 19
Elliptic part: change of formulation
From a mixed formulation on (pnh , u ) . . .
Hu +∇αsw pnh = g on Ω
divαsw u = f on Ω
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 7 / 19
Elliptic part: change of formulation
From a mixed formulation on (pnh , u ) . . .
Hu +∇αsw pnh = g on Ω
divαsw u = f on Ω
. . . to a conform formulation on pnh
− divαsw
(1
H∇αsw pnh
)= f − div
αsw
(1
Hg
)on Ω
u =1
H(g −∇αsw pnh) on Ω
under assumption 0 < H ≤ H(x) ≤ H
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 7 / 19
Elliptic part: change of formulation
From a mixed formulation on (pnh , u ) . . .
Hu +∇αsw pnh = g on Ω
divαsw u = f on Ω
. . . to a conform formulation on pnh
− divαsw
(1
H∇αsw pnh
)= f − div
αsw
(1
Hg
)on Ω
u =1
H(g −∇αsw pnh) on Ω
under assumption 0 < H ≤ H(x) ≤ H
Ani Miraçi's Master internship (summer 2017):
conforming method: easier to implement & smaller linear system
on simple 1D tests: similar accuracy as mixed formulation
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 7 / 19
How are you with combinatorics ?
If we have to analyse the convergence of each numerical method for each model. . .
METHODS
FEmixed FE
MPFA
DDFV
dG. . .
Heat equation
Stefan problem
Porous media ow
Richards equation
Incompressible Navier-Stokes. . .
PROBLEMS
each line = one analysis to perform
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 8 / 19
How are you with combinatorics ?
If we have to analyse the convergence of each numerical method for each model. . .
METHODS
FEmixed FE
MPFA
DDFV
dG. . .
Heat equation
Stefan problem
Porous media ow
Richards equation
Incompressible Navier-Stokes. . .
PROBLEMS
each line = one analysis to perform
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 8 / 19
How about including all this by some framework ?
METHODS
FEmixed FE
MPFA
DDFV
dG. . .
Heat equation
Stefan problem
Porous media ow
Richards equation
Incompressible Navier-Stokes. . .
FRAMEWORK
PROBLEMS
Objective
The framework identies a few key properties that all methodssatisfy, and that are sucient for all convergence analyses
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 9 / 19
Linear stationary diusion
−div(Λ∇u) = f in Ωu = 0 on ∂Ω
Ω open bounded in Rd
Λ : Ω→ Md(R) bounded uniformly coercive
f ∈ L2(Ω)
Idea : in the weak formulation of the PDEreplace the space and operators by the discrete ones
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 10 / 19
Linear stationary diusion
−div(Λ∇u) = f in Ωu = 0 on ∂Ω
Ω open bounded in Rd
Λ : Ω→ Md(R) bounded uniformly coercivef ∈ L2(Ω)
Idea : in the weak formulation of the PDEreplace the space and operators by the discrete ones
Weak formulation:Find u ∈ H1
0 (Ω) such that, ∀ v ∈ H10 (Ω),∫
Ω
Λ∇u · ∇v =
∫Ω
f v
Gradient scheme:Find uD ∈ XD,0 such that, ∀vD ∈ XD,0,∫
Ω
Λ∇DuD · ∇DvD =
∫Ω
f ΠDvD
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 10 / 19
Linear stationary diusion
−div(Λ∇u) = f in Ωu = 0 on ∂Ω
Ω open bounded in Rd
Λ : Ω→ Md(R) bounded uniformly coercivef ∈ L2(Ω)
Idea : in the weak formulation of the PDEreplace the space and operators by the discrete ones
Weak formulation:Find u ∈ H1
0 (Ω) such that, ∀ v ∈ H10 (Ω),∫
Ω
Λ∇u · ∇v =
∫Ω
f v
Gradient scheme:Find uD ∈ XD,0 such that, ∀vD ∈ XD,0,∫
Ω
Λ∇DuD · ∇DvD =
∫Ω
f ΠDvD
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 10 / 19
Gradient Discretisation - (GD) for homogeneous Dirichlet BC
D = (XD,0 , ΠD , ∇D)
discrete space XD,0 = Rd.o.f . (XD,0 suited to boundary
conditions)
reconstruction of function ΠD : XD,0 → L2(Ω) linear mapping
reconstruction of gradient ∇D : XD,0 → L2(Ω)d linear
mapping, such that ‖∇D · ‖L2(Ω)d is a norm on XD,0
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 11 / 19
Properties of a sequence of GDs (Dm)m∈N as m→ +∞
Coercivity (discrete Poincaré inequality)
CD = maxv∈XD,0\0
‖ΠDv‖L2‖∇Dv‖L2
CDm remains bounded
GD-Consistency (FE interpolation error)
∀ϕ ∈ H10 (Ω) , SD(ϕ) = min
v∈XD,0
( ‖ΠDv − ϕ‖L2 + ‖∇Dv −∇ϕ‖L2 )
SDm → 0
Limit-conformity (FE consistency)
∀ϕ ∈ Hdiv(Ω) , WD(ϕ) = maxu∈XD,0\0
1
‖∇Du‖L2
∣∣∣∣∫Ω
(∇Du ·ϕ + ΠDu divϕ)
∣∣∣∣WDm → 0
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 12 / 19
Application to linear stationary diusion
Weak formulation :
Find u ∈ H10 (Ω) such that, ∀ v ∈ H1
0 (Ω),∫Ω
Λ∇u · ∇v =
∫Ω
f v
Gradient scheme :
Find uD ∈ XD,0 such that, ∀vD ∈ XD,0,∫Ω
Λ∇DuD · ∇DvD =
∫Ω
f ΠDvD
Error estimate
‖ΠDuD − u‖L2 + ‖∇DuD −∇u‖L2 ≤ C (1 + CD) [SD(u) + WD(Λ∇u)]
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 13 / 19
Conforming P1 Finite Elements
On a triangular/tetrahedral mesh, V = set of vertices of the mesh
Gradient discretisation :
XD,0 := (us)s∈V : us = 0 if s ∈ ∂Ω
ΠD : XD,0 → C (Ω) ; u 7→ uh =∑s∈V
usϕs
with ϕs P1 FE shape function associated to vertex s
∇D : XD,0 → L2(Ω)d ; u 7→ ∇Du = ∇uh (piecewise constant function)I (∇Du)|K = ∇(ΠDu)|K
Poincaré inequality =⇒ coercivity
SD(ϕ) ≤ C h =⇒ GD-consistency
WD(ϕ) = 0 =⇒ limit conformity Ω
ΠDu
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 14 / 19
Summarize
GDMs in few words
a framework to study convergence analyses
replace the space and operators by the discrete ones
choose D = (XD,0, ΠD, ∇D) and ensure coercivity, consistencyand limit-conformity property
J. Droniou, R. Eymard, T. Gallouët, C. Guichard, R. Herbin, The gradient
discretisation method Springer International Publishing AG, 82, 2018,Mathématiques et Applications
J. Droniou, R. Eymard, R. Herbin, Gradient schemes: generic tools for the
numerical analysis of diusion equations (M2AN 50(3), 2016)
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 15 / 19
Summarize
GDMs in few words
a framework to study convergence analyses
replace the space and operators by the discrete ones
choose D = (XD,0, ΠD, ∇D) and ensure coercivity, consistencyand limit-conformity property
Other methods known to be GDMs
mass-lumped conforming P1 Finite Elements
some Finite Volume schemes : VAG, DDFV, SUSHI, ...
non-conforming FE method, including non-conforming Pk
Hybrid Mimetic Mixed method
Hybrid high-order methods
. . .
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 15 / 19
Elliptic part: GDMs
Elliptic problem - Ω ⊂ Rd (d = 1 or d = 2)
Find pnh : Ω→ R and u : Ω→ Rd+1 s.t.
− divαsw
(1
H∇αsw pnh
)= f − div
αsw
(1
Hg
)on Ω
Hu · ns = φ on Γn
pnh = 0 on Γd
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 16 / 19
Elliptic part: GDMs
Elliptic problem - Ω ⊂ Rd (d = 1 or d = 2)
Find pnh : Ω→ R and u : Ω→ Rd+1 s.t.
− divαsw
(1
H∇αsw pnh
)= f − div
αsw
(1
Hg
)on Ω
Hu · ns = φ on Γn
pnh = 0 on Γd
Weak formulation :
Find p ∈ H10 (Ω) such that ∀q ∈ H1
0 (Ω),∫Ω
(H∇p + p∇ζ) · (H∇q + q∇ζ) + α2pq
Hdx
=
∫Ω
f q +g1 · ∇ζ − αg2
Hq dx +
∫Ω
g1 · ∇q dx −∫
Γn
φq ds
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 16 / 19
Elliptic part: GDMs
Elliptic problem - Ω ⊂ Rd (d = 1 or d = 2)
Find pnh : Ω→ R and u : Ω→ Rd+1 s.t.
− divαsw
(1
H∇αsw pnh
)= f − div
αsw
(1
Hg
)on Ω
Hu · ns = φ on Γn
pnh = 0 on Γd
Gradient scheme :
Find p ∈ XD,0 such that ∀q ∈ XD,0,∫Ω
(H∇Dp + ΠDp∇ζ) · (H∇Dq + ΠDq∇ζ) + α2ΠDpΠDq
Hdx
=
∫Ω
f ΠDq +g1 · ∇ζ − αg2
HΠDq dx +
∫Ω
g1 · ∇Dq dx −∫
Γn
φTD,Γnq ds
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 16 / 19
Gradient Discretisation - (GD) for mixed BCs (homo. Dirichlet)
D = (XD , ΠD , ∇D , TD,Γn)
discrete space XD = XD,Γd⊕ XD,Ω,Γn direct sum of two nite
dimensional vector spaces on R
reconstruction of function ΠD : XD → L2(Ω) linear mapping
reconstruction of gradient ∇D : XD → L2(Ω)d linear
mapping, such that ‖∇D · ‖L2(Ω)d is a norm on XD,Ω,Γn
reconstruction of trace TD,Γn : XD → L2(Γn) linear mapping
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 17 / 19
Properties of a sequence of GDs (Dm)m∈N as m→ +∞
Coercivity
CD = maxv∈XD,Ω,Γn\0
(max
‖ΠDv‖L2(Ω)
‖∇Dv‖L2(Ω)d,‖TD,Γnv‖L2(Γn)
‖∇Dv‖L2(Ω)d
)CDm remains bounded
GD-Consistency
∀ϕ ∈ H1(Ω) , SD(ϕ) = minv∈XD,Ω,Γn
(‖ΠDv − ϕ‖L2(Ω) + ‖∇Dv −∇ϕ‖L2(Ω)d
)SDm → 0
Limit-conformity
∀ϕ ∈ Hdiv ,Γn(Ω) , WD(ϕ) =
maxv∈XD,Ω,Γn\0
1
‖∇Dv‖L2(Ω)d
∣∣∣∣∫Ω
(∇Dv ·ϕ + ΠDv divϕ)−∫
Γn
TD,Γnvγnφ
∣∣∣∣WDm → 0
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 18 / 19
To conclude
GDMs on the elliptic part
Gradient scheme framework includes: Conforming andNonconforming FE, some FV schemes, etc...
Deals with several BCs case-by-case
Error estimate on ‖ΠDp − p‖L2(Ω) + ‖∇Dp −∇p‖L2(Ω)d
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 19 / 19
To conclude
GDMs on the elliptic part
Gradient scheme framework includes: Conforming andNonconforming FE, some FV schemes, etc...
Deals with several BCs case-by-case
Error estimate on ‖ΠDp − p‖L2(Ω) + ‖∇Dp −∇p‖L2(Ω)d
Perspectives: Abstract GDM
Same processing of dierent BCs
Non-classic operators ∇αsw and divαsw related by a Stokes-type
formula
Can use a quadrature formula for −αp in ∇αsw p (6= −αΠDp)
Comparison between error estimate given by GDM and A-GDM ?
Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 19 / 19