On Stable Schemes for Large-Scale Non-Hydrostatic ... · Cavity test, P2 –P1 FE, standard...

On Stable Schemes for Large-ScaleNon-Hydrostatic Boussinesq Equations

J. Rafael Rodríguez GalvánWorkshop: non-hydrostatic effects in oceanography

October 15, 2018

Outline

Introduction

Constant Density Case

Stabilization of Vertical Velocity

Discontinuous Galerkin Approximation

Application to Non-Hydrostatic Large-Scale Boussinesq Equations

Section 1

Introduction

[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [3 / 39]

Large Scale Ocean Equations

The ocean: A slightly compressible fluid endowed with Coriolis and buoyancy forces

Fundamental hypothesis considered (for simplifying physical laws):

1 Boussinesq hypothesis

2 Thin aspect ratio






2 Thin aspect ratio

Boussinesq HypothesisDensity is a constant ρ? except inbuoyancy terms.






2 Thin aspect ratio

Thin aspect ratioAnisotropic (flat) domain

ε =vertical scales

horizontal scalesis small10

−3 , 10−4

E.g. Mediterranean and North Atlantic: ε ' 10−4

Then the problem is rescaled:Anisotropic domain → Isotropic domain


The adimensional domain

After vertical scaling, we obtain an isotropic domain Ω ⊂ R3:

ε =vertical scales

horizontal scales ∼ 1

Cartesian coordinates and rigid lid hypothesis (flat surface).Also free surface models might be handled by our schemes.

Non-fun

dament

al

hypoth

esis!

Vertical scaling the domain⇒ Vertical scaling the equations...


Large-Scale Boussinesq Equations (variable density)

Anisotropic equationsConservation of momentum and continuity

∂tu + u · ∇xu + v ∂zu−∆νu +1

ρ?∇xp = Fu;

ε2(∂tv + u · ∇xv + v ∂zv −∆νv

)+

1

ρ?∂zp +

ρ g

ρ?= 0

∇ · u + ∂zv = 0

Convection-diffusion of temperature and salinity + state equation (density)

∂tT + (u · ∇x)T + (v · ∂z)T − νT∆T = FT

∂tS + (u · ∇x)S + (v · ∂z)S − νS∆S = FS

ρ = ρ?(1− βT (T − T?) + βS(S − S?)

)[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [6 / 39]

Large-Scale Boussinesq Equations (variable density)

Variable density (coupling temperature and density)Conservation of momentum and continuity

∂tu + u · ∇xu + v ∂zu−∆νu +1

ρ?∇xp = Fu;

ε2(∂tv + u · ∇xv + v ∂zv −∆νv

)+

1

ρ?∂zp +

ρ g

ρ?= 0

∇ · u + ∂zv = 0

Convection-diffusion of temperature and salinity + state equation (density)

∂tT + (u · ∇x)T + (v · ∂z)T − νT∆T = FT

∂tS + (u · ∇x)S + (v · ∂z)S − νS∆S = FS

ρ = ρ?(1− βT (T − T?) + βS(S − S?)

)[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [6 / 39]

Section 2

Constant Density Case


Constant Density Non-Hydrostatic Equations

Temporarily, we introduce the constant density hypothesis

Hence we focus on momentum equations:

Quasi-Hydrostatic or Anisotropic Navier-Stokes equations

∂tu + u · ∇xu + v ∂zu−∆νu +∇xp = Fuε2(∂tv + u · ∇xv + v ∂zv −∆νv

)+ ∂zp = 0

∇ · u + ∂zv = 0

With adequate boundary conditions, e.g. νz∂zu|Γs = gs , u|Γb∪Γl = 0,

v |Γb = 0, v |Γs = 0, ∇xv · nx|Γl = 0.


Hydrostatic Navier-Stokes (or Primitive) Equations

Simplification ε = 0 leads to... Hydrostatic Navier-Stokes or Primitive Equations

∂tu + (u · ∇x)u + v∂zu−∆νu +∇xp = f in Ω,

∂zp = 0 in Ω,

∇x · u + ∂zv = 0 in Ω.

Usual approach: . . . . . . . . .

...z–integration⇒ equivalentdifferential-integral problem. . . . . . . . . .

. . . . . . . . . Pros and cons . . . . . . . . .

difficulthandling

of

non-hydrostatic case,variable density,non-sidewall boundary


Targets

Our aim:

Handle both

Hydrostatic andAnisotropic (non-hidrostatic)

...equations like standard Navier-Stokes

That means: developing an unified framework for handlinganisotropic equations for any ε ≥ 0

Not straightforward:

Usual Navier-Stokes difficultiesMixed problemLoss of elipticity when Re→ +∞

New loss of elipticity when ε→ 0


Usual FE are not valid for (Anis-NStokes) when ε→ 0

Example: P2-P1 cavity test when ε→ 0 ε = 1, ..., 10−5)

Instabilities when ε . 10−3 ..due to vertical velocity

Stream

lines

Horizo

ntal

uVe

rtical

v


A completely (mixed) differential formulationSteady linear variational formulation: find (u, v , p) ∈ U× V × P such that

ν(∇u,∇u)− (p,∇x · u) = 〈f,u〉 ∀u ∈ U,ε(∇u,∇v)− (p, ∂zv) = 0 ∀v ∈ V ,

(∇ · (u, v), p) = 0 ∀p ∈ P,

where (·, ·) is the L2(Ω) scalar product

ε→ 0, we focus on the less favorable case: hydrostatic equations (ε = 0)

ν(∇u,∇u)− (p,∇x · u) = 〈f,u〉 ∀u ∈ U,(p, ∂zv) = 0 ∀v ∈ V ,

(∇ · (u, v), p) = 0 ∀p ∈ P,

V = H1

z,0(Ω) =v ∈ L2(Ω) / ∂zv ∈ L2(Ω), v |Γs∪Γb

= 0,

Main difficulty: Loss of regularity of v


A completely (mixed) differential formulationSteady linear variational formulation: find (u, v , p) ∈ U× V × P such that

ν(∇u,∇u)− (p,∇x · u) = 〈f,u〉 ∀u ∈ U,ε(∇u,∇v)− (p, ∂zv) = 0 ∀v ∈ V ,

(∇ · (u, v), p) = 0 ∀p ∈ P,

where (·, ·) is the L2(Ω) scalar product

ε→ 0, we focus on the less favorable case: hydrostatic equations (ε = 0)

ν(∇u,∇u)− (p,∇x · u) = 〈f,u〉 ∀u ∈ U,(p, ∂zv) = 0 ∀v ∈ V ,

(∇ · (u, v), p) = 0 ∀p ∈ P,

V = H1

z,0(Ω) =v ∈ L2(Ω) / ∂zv ∈ L2(Ω) , v |Γs∪Γb

= 0,

Main difficulty: Loss of regularity of v


Analysis of Hydrostatic Stokes equations

Well-posedness (∃! of solution + stability estimates)of the hydrostatic mixed problem can be shown1 using:

1 The standard (Stokes) LBB inf-sup condition:

βp‖p‖≤ sup0 6=(u,v)∈U×V

(∇ · (u, v), p)

‖(∇u, ∂zv)‖∀p ∈ P(IS)P

2 But also an additional “hydrostatic” inf-sup condition!!

βv‖∂zv‖≤ sup06=p∈P

(∂zv , p)

‖p‖∀v ∈ V(IS)V

1[Azé96], [GGRG15a][J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [13 / 39]

The discrete FE case

Uh ⊂ U, Vh ⊂ V , Ph ⊂ P conforming FE spacesDiscrete problem: find (uh, vh, ph) ∈ Uh × Vh × Ph such that

ν(∇uh,∇uh)− (ph,∇x · uh) = (f,uh) ∀uh ∈ Uh

(ph, ∂zvh) = 0 ∀vh ∈ Vh

(∇ · (uh, vh), ph) = 0 ∀ph ∈ Ph

(IS)Ph , (IS)Vh : discrete versions of (IS)P , (IS)V

Theorem ([Azé96], [GGRG15a])Problem iswell-posed

for (Uh,Vh)− Ph⇔ both (IS)Vh and (IS)Vh hold

But usual Stokes FE do not verify (IS)Vh !!That is the reason for instability of P2-P1, P1,b-P1, etc. [GGRG15a]


The discrete FE case

Uh ⊂ U, Vh ⊂ V , Ph ⊂ P conforming FE spacesDiscrete problem: find (uh, vh, ph) ∈ Uh × Vh × Ph such that

ν(∇uh,∇uh)− (ph,∇x · uh) = (f,uh) ∀uh ∈ Uh

(ph, ∂zvh) = 0 ∀vh ∈ Vh

(∇ · (uh, vh), ph) = 0 ∀ph ∈ Ph

(IS)Ph , (IS)Vh : discrete versions of (IS)P , (IS)V

Theorem ([Azé96], [GGRG15a])Problem iswell-posed

for (Uh,Vh)− Ph⇔ both (IS)Vh and (IS)Vh hold

But usual Stokes FE do not verify (IS)Vh !!

That is the reason for instability of P2-P1, P1,b-P1, etc. [GGRG15a]


Section 3

Stabilization of Vertical Velocity


Our purpose

To reformulate Hydrostatic Stokes equations

in order to avoid the hydrostatic restriction (IS)Vh


Reformulated system

Find (u, v , p) ∈ U× V × P such that

ν(∇u,∇u)− (p,∇x · u) = 〈f,u〉 ∀u ∈ U

ν(∇ · (u, v), ∂zv) − (p, ∂zv) = 0 ∀v ∈ V

(∇ · (u, v), p) = 0 ∀p ∈ P

Obtained by adding the consistent? term ν(∇ · (u, v), ∂zv)

? it vanishes, due toincompressibility and∂zV ∈ P


Discrete versionFind FE functions uh ∈ Uh , vh ∈ Vh and ph ∈ Ph such that

ν(∇uh,∇uh)− (ph,∇x · uh) = 〈f,uh〉 ∀uh ∈ Uh

ν(∇ · (uh, vh), ∂zvh) − (ph, ∂zvh) = 0 ∀vh ∈ Vh

(∇ · (uh, vh), ph) = 0, ∀ph ∈ Ph

Not equivalent to original Hydrostatic problem, because, ingeneral,

ν(∇ · (uh, vh), ∂zvh) 6= 0

(since ∂zVh 6⊂ Ph for most FE spaces)

Anyway, we can showa) well-posedness and

b) convergence to continuous Hydrostatic equations

...imposing only (IS)Ph (even if (IS)Vh does not hold)[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [18 / 39]

a) Well-posedness

Theorem (Well-posedness)Let Uh ⊂ U, Vh ⊂ V and Ph ⊂ P be families of FE

satisfying the inf-sup condition (IS)Ph .

The stabilized scheme has a unique solution

(uh, vh, ph) ∈ Uh × Vh × Ph

which satisfies the following stability estimates:

‖∇uh‖2 + ‖∂zvh‖2 ≤4

ν2‖f‖2U′ , ‖ph‖ ≤

5

γp‖f‖U′ ,

where γp is the constant in (IS)Ph .


b) Convergence and Error Orders

Applying standard reasoning from Mixed FE we can show

convergence results similar to Stokes equations

Example: ifuh, vh ∼ Prph ∼ Pr−1

then:

‖∇(u−Uh)‖0 + ‖∂z(v − vh)‖0 + ‖p − ph‖0 ≤

Chr(‖(u, v)‖Hr+1(Ω)d + ‖p‖Hr (Ω)

).

In particular (in energy norm): order O(h2) for P2 -P1 if (u, v) ∈ H3(Ω) and p ∈ H2(Ω) order O(h) for P1,b -P1 if (u, v) ∈ H2(Ω) and p ∈ H1(Ω)


P2 − P1

P1,b − P1[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [21 / 39]

Remarks

1 All results are valid for the

Non-Hydrostatic case

2 In our numerical tests, we are not worried about ε

(Hydrostatic and Non-Hydrostatic cases handled in the same way)

See [GGRG15b] for details


Cavity test in a domain with an internal talus

Cavity test, P2–P1 FE, standard unstructured mesh (with adaptivity)

Recirculation and hydrostatic pressure: qualitatively correct

Small instabilities due to singularity in bottom


Pressure in convex domain with no sidewallCavity test, P2–P1 FE

Refined mesh

Pressure v–stabilized scheme ∂zp–regularized scheme.The ∂zp–regularized scheme seems best for handling singularitiesEasy programming! (we used FreeFem++)


Realistic 3D meshes in Gibraltar Strait

We exploited the facilities of our schemes for 3D tests inunstructured meshesReal Earth data:

Coast lines from naturalearthdata.comBathymetry from the U.S.A. National Geophysical Data Center(ETOPO2v2).


Academic 3D cavity test in Gibraltar Strait

3D cavity test in the Gibraltar Strait


Section 4

Discontinuous Galerkin Approximation


Idea

Facts:Pr/Pr velocity/pressure FE verify (IS)Vh

But they are not (IS)Ph stable

Although they are stable for SIP DG Stokes approximations!

Pr/Pr are stable for SIP DG for Anisotropic Stokesequations?


DG Bilinear Form for Anisotropic Stokes

Order k polynomials for velocity and pressure:

Anisotropic velocity bilinear form:

Horizontal velocity: usual Stokes SIP DG approximationVertical velocity: ε–dependent bilinear form

ah(wh,wh) =( d−1∑

i=1

asip,ηh (ui , ui ) + aεh(vh, vh)),

whereaεh(vh, vh) =

ε asip,ηh (vh, vh) + (1− ε)svh (vh, vh),

svh (vh, vh) =∑e∈Eh

1he

∫e

([[vhnz ]] [[vhnz ]]

).

[?]


DG Bilinear Form for Anisotropic Stokes

Order k polynomials for velocity and pressure:

Anisotropic velocity bilinear form:

Horizontal velocity: usual Stokes SIP DG approximationVertical velocity: ε–dependent bilinear form

ah(wh,wh) =( d−1∑

i=1

asip,ηh (ui , ui ) + aεh(vh, vh)),

whereaεh(vh, vh) =

ε asip,ηh (vh, vh) + (1− ε)svh (vh, vh),

svh (vh, vh) =∑e∈Eh

1he

∫e

([[vhnz ]] [[vhnz ]]

).

[?][J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [29 / 39]

Coercivity

aanish (wh,wh) ≥ C (η)(‖uh‖2sip + ε‖vh‖2sip

)+ (1− ε)|vh|2V

Stability for ‖ph‖0

β ‖ph‖0 ≤ supwh∈Wh\0

bStoh (wh, ph)

‖wh‖sip+ |ph|P , ∀ph ∈ Ph.

Stability for ∂z,hvh

‖∂z,hvh − 〈∂z,hvh〉Ω‖0 = supph∈Ph

∫Ω ph ∂z,h vh

‖ph‖0,

where 〈∂z,hvh〉Ω is the mean of ∂z,hvh in Ω

⇒ Stability and well-posedness of SIP DG Anisotropic Stokes(no need of vertical stabilizing)


SIP DG Numerical Test

Figure 2. DG Non-Hydrostatic Cavity test (ε = 10−8).


Section 5

Application to Non-Hydrostatic Large-ScaleBoussinesq Equations


Euler method

∂tu + u · ∇xu + v ∂zu−∆νu +1ρ?∇xp =

ε2(∂tv + u · ∇xv + v ∂zv −∆νv

)+

1ρ?∂zp +

ρ g

ρ?= 0

∇ · u + ∂zv = 0

∂tT + (u · ∇x)T + (v · ∂z)T − νT∆T = FT

∂tS + (u · ∇x)S + (v · ∂z)S − νS∆S = FS

ρ = ρ?(1− βT (T − T?) + βS(S − S?)

)Euler semi-implicit scheme:

Step 1. Calculate temperature, salinity and then density (decouplingfrom momentum equations)

Step 2. Use density to obtain velocity and pressure (solveAnisotropic Navier-Stokes using previous techniques)


Cavity test + temperature

ε = 0, P2/P2 -P1 elements, “random” unstructured meshFlow: wind traction on surface (left to right)Temperature: heat transfer from surface

Stream

lines

Tempe

rature


Gibraltar strait 2D flow + temperature + salinityStream

lines

Salinity

Tempe

rature


Conclusions

Unified framework for approximation of Non-HydrostaticNavier-Stokes and Primitive Equations of the Ocean

Idea: stabilizing vertical velocity makes possible:

The use of standard FE tools and techniquesWith optimal (Stokes-like) error orders.

Also exploring Discontinuous Galerkin approximations

Straightforward extension to Boussinesq (variable density)equations


Thankyou!

Bibliography I

P. Azérad. Analyse des Ãľquations de Navier-Stokes en bassin peuprofond et de l’Ãľquation de transport. PhD thesis, NeuchÃćtel,1996.F. Brezzi and M. Fortin.Mixed and Hybrid Finite Element Methods.Springer-Verlag, New-York, 1991.F. Guillén-González and J.R. Rodríguez-Galván. Analysis of thehydrostatic stokes problem and finite-element approximation inunstructured meshes. Numerische Mathematik, 2015.DOI:10.1007/s00211-014-0663-8.F. Guillén-González and J.R. Rodríguez-Galván. Stabilized schemesfor the hydrostatic stokes equations. SIAM Journal on NumericalAnalysis, 2015. To appear.

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