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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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [4 / 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
Boussinesq HypothesisDensity is a constant ρ? except inbuoyancy terms.
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [4 / 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
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [4 / 39]
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...
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [5 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [7 / 39]
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.
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [8 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [9 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [10 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [11 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [12 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [12 / 39]
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]
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [14 / 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]
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [14 / 39]
Section 3
Stabilization of Vertical Velocity
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [15 / 39]
Our purpose
To reformulate Hydrostatic Stokes equations
in order to avoid the hydrostatic restriction (IS)Vh
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [16 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [17 / 39]
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 .
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [19 / 39]
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)
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [20 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [22 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [23 / 39]
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++)
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [24 / 39]
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).
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [25 / 39]
Academic 3D cavity test in Gibraltar Strait
3D cavity test in the Gibraltar Strait
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [26 / 39]
Section 4
Discontinuous Galerkin Approximation
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [27 / 39]
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?
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [28 / 39]
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]
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)
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [30 / 39]
SIP DG Numerical Test
Figure 2. DG Non-Hydrostatic Cavity test (ε = 10−8).
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [31 / 39]
Section 5
Application to Non-Hydrostatic Large-ScaleBoussinesq Equations
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [32 / 39]
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)
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [33 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [34 / 39]
Gibraltar strait 2D flow + temperature + salinityStream
lines
Salinity
Tempe
rature
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [35 / 39]
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
[J. R. Rguez. Galván. Stable Schemes Non-Hydrostatic Large-Scale] [36 / 39]
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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