Non-hydrostatic shallow water model and Gradient ... · Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 1 / 19. Outline 1 Dispersive e ects modelling

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)


Free-surface ow modelling

Free-surfaceincompressibleEuler equations

Shallow water

equations

Hydrostatic pressure

Homogeneous velocity

Shallow water assumption


Free-surface ow modelling

Free-surfaceincompressibleEuler equations

Shallow water

equations

hhhhhhhhhHydrostatic pressure

Homogeneous velocity

Shallow water assumption


Emphasis of non-hydrostatic eects

M.-W. Dingemans, Wave propagation over uneven bottoms (Adv. Ser.Ocean Eng., 1997)


Emphasis of non-hydrostatic eects


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)




water height H

velocity u = (v ,w)


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




water height H

velocity u = (v ,w)


α = 2 (α = 32:

Serre-Green-Naghdi)

∂H

∂t+ ∇ · (Hu) = 0

∂(Hu)

∂t+ ∇ · (Hu⊗ u) + ∇

(gH2

2


divαsw u = 0

Numerical strategy: Time splitting (projection/correction) Hyperbolic solver /Dispersive solver




water height H

velocity u = (v ,w)


α = 2 (α = 32:

Serre-Green-Naghdi)

∂H

∂t+ ∇ · (Hu) = 0

∂(Hu)

∂t+ ∇ · (Hu⊗ u) + ∇

(gH2

2


divαsw u = 0

Numerical strategy: Time splitting (projection/correction) Hyperbolic solver /Dispersive solver


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)


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


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


Elliptic part: change of formulation

From a mixed formulation on (pnh , u ) . . .

Hu +∇αsw pnh = g on Ω

divαsw u = f 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





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


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


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


PROBLEMS

each line = one analysis to perform


How about including all this by some framework ?

METHODS

FEmixed FE

MPFA

DDFV

dG. . .

Heat equation

Stefan problem

Porous media ow

Richards equation


FRAMEWORK

PROBLEMS

Objective

The framework identies a few key properties that all methodssatisfy, and that are sucient for all convergence analyses


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





Λ : Ω→ Md(R) bounded uniformly coercivef ∈ L2(Ω)


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





Λ : Ω→ Md(R) bounded uniformly coercivef ∈ L2(Ω)


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


∫Ω

f ΠDvD


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


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


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


∫Ω

f ΠDvD

Error estimate

‖ΠDuD − u‖L2 + ‖∇DuD −∇u‖L2 ≤ C (1 + CD) [SD(u) + WD(Λ∇u)]


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


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)


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

. . .


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


pnh = 0 on Γd


Elliptic part: GDMs



− divαsw

(1

H∇αsw pnh

)= f − div

αsw

(1

Hg

)on Ω


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


Elliptic part: GDMs



− divαsw

(1

H∇αsw pnh

)= f − div

αsw

(1

Hg

)on Ω


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


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


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


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


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 ?


Documents

Non-hydrostatic shallow water model and Gradient ... · Virgile DUBOS (LJLL - ANGE) Non-hydrostatic SW model & GDMs October 16, 2018 1 / 19. Outline 1 Dispersive e ects modelling