A class of finite volume schemes for the 2D shallow water equations - - PowerPoint PPT Presentation

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A class of finite volume schemes for the 2D shallow water equations - - PowerPoint PPT Presentation

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A class of finite volume schemes for the 2D shallow water equations with Coriolis force E. Audusse 1 , V. Dubos 2 , A. Duran 3 , N. Gaveau 4 , Y. Nasseri 5 , Y.Penel 2 1 Universit Paris 13, LAGA, CNRS, UMR 7539, Institut Galile Villetaneuse,

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A class of finite volume schemes for the 2D shallow water equations with Coriolis force

  • E. Audusse1, V. Dubos2, A. Duran3, N. Gaveau4, Y. Nasseri5,

Y.Penel2

1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inria–Paris, Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598,

Laboratoire Jacques-Louis Lions, France.

3 Institut Camille Jordan, Université Claude Bernard Lyon 1, France. 4 Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, France. 5 Institut de Mathématique de Marseille, Aix-Marseille Université, France.

August 22, 2019 CEMRACS 2019

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 1 / 17

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SW equations with Coriolis source term

Ω being an open bounded domain of R2, flat bottom and T > 0.            ∂th +∇·(hu) =

in Ω×(0,T),

∂t(hu)+∇·(hu ⊗ u)+ g h∇h = −ωhu⊥,

in Ω×(0,T), u · n

=

  • n ∂Ω×(0,T),

h(x,0)

=

h0 in Ω, u(x,0)

=

u0 in Ω. Energy balance equation:

∂tE +∇·((1

2|u|2 + gh)hu) = 0, with E = 1 2 g h2 + 1 2 h|u|2.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 2 / 17

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Linearised SW equations with Coriolis source term

Linear equations around u0 = 0 and h0>0:

∂th + h0 ∇· u =

in Ω×(0,T),

∂tu + g ∇h = −ωu⊥,

in Ω×(0,T). Energy balance equation:

∂tE +∇·(E u) = 0, with E = 1

2 g h2 + 1 2 |u|2 Geostrophic equilibrium: g ∇h +ωu⊥ = 0.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 3 / 17

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Linearised SW equations with Coriolis source term

Geostrophic equilibrium: g ∇h +ωu⊥ = 0.

Source : M. H. Do, Mathematical analysis of finite volume schemes for the simulation of quasi-geostrophic flows at low Froude number, 2017.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 3 / 17

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Outline

Aim: 2D entropic scheme, consistent kernel with geostrophic equilibrium.

1

State of art Inaccuracy of the classic Godunov scheme Linearised SW with Coriolis source term Energy dissipative scheme for SW

2

Collocated semi-discrete scheme Modified equations Non-linear equations Linear equations

3

Mixed semi-discrete scheme Non-linear scheme Linear scheme

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 4 / 17

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Inaccuracy of the classical Godunov scheme

Modified equations:

   ∂tr + a⋆∇· u − κra⋆(∆x∂2

xr +∆y∂2 yr)

= ∂tu + a⋆∇xr − κua⋆∆x∂2

xu

= ωv ∂tv + a⋆∇yr − κua⋆∆y∂2

yv

= −ωu

Source : E. Audusse, M. H. Do, P . Omnes, and Y. Penel [1]

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 5 / 17

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Linearised SW with Coriolis source term

[1] E. Audusse, M. H. Do, P . Omnes, and Y. Penel, Analysis of modified godunov type schemes for the two-dimensional linear wave equation with coriolis source term on cartesian meshes, JCP , 2018.

Cell-centered semi-discrete scheme

dri,j(t)

dt

+ a∗[∇h · uh]i,j −νr

  • ∇h ·
  • ∇hrh + ω

a∗ u⊥ h

  • i,j

= 0,

dui,j(t)

dt

+ a∗[∇hrh]i,j −νu [∇h(∇h · uh)]i,j = −ωu⊥

i,j.

preserves geostrophic equilibrium,

(∇hrh + ω

a∗ u⊥ h = 0) =

⇒ (∇h · uh = 0),

full discrete energy dissipation (νr = 0), vertex-based version.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 6 / 17

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Energy dissipative scheme for SW

[2] F . Couderc, A. Duran and J.-P . Vila, An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification, JCP , 2017.

Colocated explicit version

hn+1

K

= hn

K − ∆t

mK ∑

e∈∂K

F n

e ·

ne,K me , hn+1

K

un+1

K

= hn

K un K − ∆t

mK ∑

e∈∂K

  • un

K (F n e ·

ne,K )+

− un

Ke (F n e ·

ne,K )− me

− ∆t

mK ghn

K ∑ e∈∂K

h∗,n

e

  • ne,K me .
  • he = 1

2 (hK + hKe)

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 7 / 17

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Energy dissipative scheme for SW

Colocated explicit version

hn+1

K

= hn

K − ∆t

mK ∑

e∈∂K

F n

e ·

ne,K me , hn+1

K

un+1

K

= hn

K un K − ∆t

mK ∑

e∈∂K

  • un

K (F n e ·

ne,K )+

− un

Ke (F n e ·

ne,K )− me

− ∆t

mK ghn

K ∑ e∈∂K

h∗,n

e

  • ne,K me .
  • he = 1

2 (hK + hKe)

F n

e = (hu)n e −γ g Πn e

, Πn

e −

→ ∇e[h] =

  • hn

Ke − hn K

  • ne,K .

h∗

e = hn e −αΛn e

, Λn

e −

→ ∇e ·[hu] =

  • hun

Ke − hun K

  • ·

ne,K .

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 7 / 17

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Proposed scheme

Semi-discrete scheme

  • dhK

dt

= −∇K ·F

dhK uK

dt

= −∇up

K ·(u,F )− ghK∇K h −ωhK u⊥ K +ωΠ⊥ K

where :

∇up

K ·(u,F ) =

∑

i∈{e,w,n,s}

|i| |K|

  • uK(FK,i · eK,Ki)+ + ui(FK,i · eK,Ki)−

∇K ·F = FK,e −FK,w

2∆x

· e1 + FK,n −FK,s

2∆y

· e2 ∇K h = he − hw

2∆x e1 + hn − hs 2∆y e2

FK,i = 1

2(hK uK +hiui −ΠK −Πi)

ΠK = γ∆t hk(g∇K h +ωu⊥

K )

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 8 / 17

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Modified equations associated scheme

Semi-discrete scheme

  • d

dthk

= −∇K · huK +∇K ·ΠK

d

dthK uK

= −∇up

K ·(u,F )− ghK∇K h −ω hu⊥ K +ωΠ⊥ K

with ΠK = γ∆t hk(g∇K h +ωu⊥

K )

Modified non linear equations

  • ∂th

= −∇· hu +∇·Π ∂t(hu) = −∇·(F ⊗ u)− g h ∇h −ω hu⊥ +ωΠ⊥

where : Π = γ∆t (g h∇h +ωhu⊥). Mechanic energy balance of the modified equations :

∂t

  • 1

2(gh2 + hu2)

  • +∇·
  • (gh + 1

2|u|2)hu

  • = −γ∆t
  • g h∇h +ωhu⊥

2.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 9 / 17

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Energy of the scheme

d dt

  • ∑

K∈T

1 2(gh2

K + hK|u|2 K)

  • = ∑

K∈T

|K|

  • ghK∇K ·Π+ωΠ⊥

K · uK + RK

  • where

RK = 1 2 1

|K| ∑

e∈∂K

|e|uKe − uK2(Fe · ne,K)− ≤ 0

∑

K∈T

(ghK∇K ·Π+ωΠ⊥

K uK) = − ∑ K∈T

((g∇K h +ωu⊥

K )·ΠK) ≤ 0

ΠK = γ∆t (g hK ∇K h +ωhu⊥).thanks to the grad-div duality,

preserved by our discretisations

Energy decreasing property

d dt

  • ∑

K∈T

1 2(gh2

K + hK|u|2 K)

  • ≤ 0
  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 10 / 17

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Linearisation of the scheme

Semi-discrete linearised scheme around h0 > 0 and u0 = 0

d dt hK + ∇K ·(h0 uK −ΠK)

=

0, h0 d dt uK + g h0 ∇K h

= −ω(h0 uK −ΠK)⊥.

where : ΠK = h0 γ∆t (g∇K h +ωu⊥

K ).

Remark : Very similar to Audusse and al [1]. Modified equations associated:

∂th +∇·F =

0,

∂th0u + g h0 ∇h = −ωF ⊥,

where: F = h0u −γ ∆t h0 (g ∇h +ω u⊥). Geostrophic equlibrium: g ∇h +ωu⊥ = 0

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 11 / 17

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The kernel

The discrete kernel associated to the geostrophic equilibrium is :

K k = {(hk,uK) | g∇K h +ωu⊥

K = 0}

and

Properties of the kernel

is consistent with the continuous one, is preserved by the linearised version of our scheme. That is : if (hk(0),uk(0)) ∈ K k then (hk(t),uk(t)) ∈ K k, ∀t ≥ 0. Remark : The geostrophic equilibrium is, by definition, not an equilibrium for the non linear shallow water model.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 12 / 17

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Mixed semi-discrete scheme

Non-linear scheme

  

d dt hK +∇K ·(F )

=

0,

d dt (hu)K +∇up K ·(uF )+ ghK ∇K h

= −ωhK u⊥

K +ω ∑ e∈∂K

1 2 Π⊥

e

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 13 / 17

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Mixed semi-discrete scheme

Non-linear scheme

  

d dt hK +∇K ·(F )

=

0,

d dt (hu)K +∇up K ·(uF )+ ghK ∇K h

= −ωhK u⊥

K +ω ∑ e∈∂K

1 2 Π⊥

e

F = hu −γ∆t h (g ∇h +ωu⊥)

Πe = γ∆t he (g ∇eh +ωu⊥

e )

∇eh = |e| |K| (hKe − hK)e1, with e = K|Ke

he = 1

2(hKe + hK) and ue = 1 2(uKe + uK)

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 13 / 17

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Mixed semi-discrete scheme

Non-linear scheme Discrete mechanic energy:

E(t) = ∑

K∈T

|K| 1

2(gh2

K + hK|u|2 K).

Decreasing of the mechanic energy

E′(t) ≤ 0.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 14 / 17

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Mixed semi-discrete scheme

Linear scheme

  

d dt hK +∇K ·(F )

=

0, h0

d dt uK + gh0 ∇K h

= −ωh0 u⊥

K +ω ∑ e∈∂K

1 2 Π⊥

e

F = h0u −γ∆t h0 (g ∇h +ωu⊥)

Πe = γ∆t h0 (g ∇eh +ωu⊥

e )

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 15 / 17

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Mixed semi-discrete scheme

Linear scheme

Decreasing of the mechanic energy

E′(t) ≤ 0.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 16 / 17

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Mixed semi-discrete scheme

Linear scheme

Decreasing of the mechanic energy

E′(t) ≤ 0.

Discrete kernel: KK = {(hK,uK) | ΠK = 0}.

Well balanced property ?

(hK,uK)(t = 0) ∈ KK ⇒ (hK,uK)(t > 0) ∈ KK,

Since ΠK = 0 ⇒ Πe = 0. We have to choose between dissipative energy and preservation of the geostrophic equilibrium, cannot have both.

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 16 / 17

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Conclusion

Results

Several schemes for non-linear and linear shallow water with Coriolis force Semi-discrete energy stability Preservation of the geostrophic equilibrium for the collocated semi-discrete scheme Staggered semi-discrete scheme

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 17 / 17

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Conclusion

Results

Several schemes for non-linear and linear shallow water with Coriolis force Semi-discrete energy stability Preservation of the geostrophic equilibrium for the collocated semi-discrete scheme Staggered semi-discrete scheme

Perspectives

Time discretisation of Coriolis source term (work in progress ) Fully discrete energy inequality Addition of a pressure correction Numerical simulation (work in progress )

  • V. Dubos, N. Gaveau, Y. Nasseri (1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr

FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 17 / 17