Présentation des schémas d’advection (moment et traceurs) et opérateurs de viscosité/diffusion présents dans CROCO

Equations primitives Reynolds averaged

Schémas d’advection

Rappels : ordre d’un schéma

But : trouver une approximation des dérivées partielles => séries de taylor au point x

Ordre 4

Ordre 3

Ordre 2

Ordre 1

Ordre 1

Un « bon » schéma

- Stable, CFL=- Précis

- Peu dispersif- Peu diffusif

- Peu couteux

density anomaly (kilogram meter-3)

x-dimension of the grid

S-co

ordi

nate

at R

HO-p

oint

s

Gravitational Adjustment ExampleRange of density anomaly: 15.7401 to 18.2599 kilogram meter-3Range of x-dimension of the grid: -0.279297 to 286.279Range of S-coordinate at RHO-points: -0.999023 to -0.000976562Current time since initialization: 492 secondCurrent y-dimension of the grid: 1Frame 42 in File khinst_his.nc

pmar

ches

Tue

Oct

2 1

8:16

:05

2018

density anomaly (kilogram meter-3)

x-dimension of the grid

S-co

ordi

nate

at R

HO-p

oint

s

Gravitational Adjustment ExampleRange of density anomaly: 15.74 to 18.26 kilogram meter-3Range of x-dimension of the grid: -0.279297 to 286.279Range of S-coordinate at RHO-points: -0.999023 to -0.000976562Current time since initialization: 492 secondCurrent y-dimension of the grid: 1Frame 42 in File khinst_his_512_TS_VADV_SPLINES.nc

pmar

ches

Wed

Oct

3 1

3:42

:04

2018

density anomaly (kilogram meter-3)

x-dimension of the grid

S-co

ordi

nate

at R

HO-p

oint

s

Gravitational Adjustment ExampleRange of density anomaly: 15.7401 to 18.2599 kilogram meter-3Range of x-dimension of the grid: -0.279297 to 286.279Range of S-coordinate at RHO-points: -0.999023 to -0.000976562Current time since initialization: 492 secondCurrent y-dimension of the grid: 1Frame 42 in File khinst_his_WENO_512.nc

pmar

ches

Tue

Oct

2 1

8:16

:15

2018

�t <�x

Vmax<latexit sha1_base64="UQCnLyA6luFHvE3kPJvwAomaokE=">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</latexit><latexit sha1_base64="lG3N0I9yujXNNTir05NAP42aDl0=">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</latexit><latexit sha1_base64="lG3N0I9yujXNNTir05NAP42aDl0=">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</latexit><latexit sha1_base64="CAbR8W7AIqN5y2N+J51dAOteO9w=">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</latexit>

Options pour l’advection

Equation horizontal vertical

Moment 3D (UV_ADV)(+ eq W_ en NBQ)

UV_HADV_TVDUV_HADV_C2

UV_HADV_UP3 UV_HADV_C4

UV_HADV_UP5UV_HADV_C6

UV_HADV_WENO5

UV_VADV_TVDUV_VADV_C2

UV_VADV_SPLINES UV_VADV_WENO5

Traceurs

TS_HADV_UP3 TS_HADV_RSUP3

TS_HADV_C4TS_HADV_WENO5

TS_HADV_UP5TS_HADV_C6

TS_HADV_RSUP5

TS_VADV_TVDTS_VADV_C2

TS_VADV_SPLINESTS_VADV_AKIMA TS_VADV_WENO5

Moment 2D M2_HADV_UP3 M2_HADV_C2

Le problème d’advection : un problème d’interpolation

qi,j

ui+ 12 ,j

ui� 12 ,j

vi,j� 12

vi,j+ 12

Besoin d’evaluer qaux interfaces

x

y

qi,j

ui+ 12 ,j+1

vi,j+ 12

vi,j� 12

qi+1,j

vi+1,j+ 12

vi+1,j� 12

Besoin d’evaluer u et vau centre et aux coins

des mailles

x

y

MomentTraceurs

Advection horizontale

HADV_C2, HADV_UP3, HADV_C4,HADV_UP5, HADV_C6eqi� 1

2

qi�1 qiqi�2qi�3 qi+1 qi+2

ui� 12

Advection horizontale

HADV_RSUP3,HADV_RSUP5eqi� 1

2

qi�1 qiqi�2qi�3 qi+1 qi+2

ui� 12

=> Schémas Split-UP3 et Split-UP5

Propriétés des schémas linéaires

0 ⇡4

⇡2

3⇡4

⇡

k�x

0.0

0.2

0.4

0.6

0.8

1.0

Am

plifi

catio

nfa

ctor

C2UP3C4UP5C6

50�x 10�x 4�x 2�xwavelength

0 ⇡4

⇡2

3⇡4

⇡

k�x

0.0

0.2

0.4

0.6

0.8

1.0

Pha

seer

ror

C2UP3C4UP5C6

50�x 10�x 4�x 2�xwavelength

q = q0eikx�!t

<latexit sha1_base64="9LJ0RBlv2t1ATsSaieIsUR3h6lQ=">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</latexit><latexit sha1_base64="/Zl5mpySYrRVH6jMrIgG2pe0BPk=">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</latexit><latexit sha1_base64="/Zl5mpySYrRVH6jMrIgG2pe0BPk=">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</latexit><latexit sha1_base64="4idw3p8SFUD0DlfHjOz7jJXiWrE=">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</latexit>

qn = (A)nq0, we want A <= 1<latexit sha1_base64="BfxShEPXz2ACyc9SCqvi38DYOhw=">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</latexit><latexit sha1_base64="kU8fpnfkxKmOOM1gNGaUpUcAS1w=">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</latexit><latexit sha1_base64="kU8fpnfkxKmOOM1gNGaUpUcAS1w=">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</latexit><latexit sha1_base64="5S+crkDnuhkYXL8cycukYpx20bo=">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</latexit>

Interprétation différences finies vs volumes finis

Schéma UP3 classique

Pour annuler l’erreur d’ordre 2

Schéma UP3 au sens des volumes finis (ex Leonard 95)

xixi�1xi�2

xi� 12

qi

qi�2

qi�1

eqk� 12

Schémas WENO (e.g Acker et al, 2016), TS_HADV_WENO5

xixi�1xi�2

xi� 12

qi

qi�2

qi�1

eqk� 12

xi�3

qi�3

xi+1

qi+1

Pondération non linéaire entre 3 évaluations de la valeur d’interface

1. Convexité2. Propriété ENO (essentially non-oscillatory)3. 5e ordre si q(x) est lisse

1. Propriété ENO (il satisfait une contrainte Totale Variation Bounded)

2. Mais il ne préserve pas la monotonie !!!!

=> néanmoins : choix privilégié pour les traceurs biogéochmiques

x

q

8N, 9M,N�1X

j=1

|qj+1 � qj | M

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BIO_HADV_WENO5

Schémas TVD, UV_HADV_TVD

Pondération entre un upwind et un C2

1. Total Variation Diminishing => monotone2. Mais est d’ordre 1

eqi� 12

qi�1 qiqi�2qi�3 qi+1 qi+2

ui� 12

eqTV Di� 1

2= (1� ⇠)eqC2

i� 12+ ⇠eqUPW

i� 12

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⇠ = f(r)<latexit sha1_base64="7P6WH/qoHk2cWPtmYRT9//KZNnE=">AAACy3icjVHLSsNAFD2Nr1pfVZduQotQEUriRjdC0Y0boYJ9QFskSad1ME3CZCLW6tIfcKsL/0r8A/0L70xTUIvohCRnzj3nztx73cjnsbSst4wxMzs3v5BdzC0tr6yu5dc36nGYCI/VvNAPRdN1YubzgNUklz5rRoI5A9dnDffqWMUb10zEPAzO5TBinYHTD3iPe44kqtm+4Ye9kti5yBetsqWXOQ3sFBQrhfbuC4BqmH9FG12E8JBgAIYAkrAPBzE9LdiwEBHXwYg4QYjrOMM9cuRNSMVI4RB7Rd8+7VopG9Be5Yy126NTfHoFOU1skycknSCsTjN1PNGZFftb7pHOqe42pL+b5hoQK3FJ7F++ifK/PlWLRA8HugZONUWaUdV5aZZEd0Xd3PxSlaQMEXEKdykuCHvaOemzqT2xrl311tHxd61UrNp7qTbBh7olDdj+Oc5pUN8r21bZPqNJH2G8sthCASWa5z4qOEEVNT3HRzzh2Tg1YuPWuBtLjUzq2cS3ZTx8Agdik3M=</latexit><latexit sha1_base64="KmuWTX92rh9PDOpE0t9lrZnkVJ4=">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</latexit><latexit sha1_base64="KmuWTX92rh9PDOpE0t9lrZnkVJ4=">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</latexit><latexit sha1_base64="auQVDjaZ/D9sO9UzgT8B3xsZSeg=">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</latexit>

ui� 12> 0 : eqTV D

i� 12

= qi�1 ; r =qi � qi�1

qi+1 � qi<latexit sha1_base64="4C2jDiiKaMy1oHD0MR6RLN98nOA=">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</latexit><latexit 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Termes non-linéaires (rhs3d.F)Formulation à la Lilly (1965)

UV_HADV_UP3

avec le choix de la direction de l’upwind fait en fonction de :

qi,j

ui+ 12 ,j+1

vi,j+ 12

vi,j� 12

qi+1,j

vi+1,j+ 12

vi+1,j� 12

Besoin d’evaluer u et vau centre et aux coins

des mailles

x

y

Schémas d’advection verticale : VADV_SPLINES

Les flux sont obtenus comme solution du problème tri-diagonal:

0 ⇡4

⇡2

3⇡4

⇡

k�x

0.0

0.2

0.4

0.6

0.8

1.0

Am

plifi

catio

nfa

ctor

SplinesUP5C6

50�x 10�x 4�x 2�xwavelength

0 ⇡4

⇡2

3⇡4

⇡

k�x

0.0

0.2

0.4

0.6

0.8

1.0

Pha

seer

ror

SplinesUP5C6

50�x 10�x 4�x 2�xwavelength

hk

hk+1

hk�1

⇠

�hk/2

+hk/2

qk

qk+1

qk�1

eqk� 12

eqk+ 12

q(⇠)

q(⇠) = qk +eqk+ 1

2� eqk� 1

2

hk⇠

+ 6

eqk+ 1

2+ eqk� 1

2

2� qk

!✓⇠2

h2k

� 1

12

◆

Formulation semi-implicite

V_ADV_IMP

Idée : séparer la vitesse verticale en une contribution explicite et une contributionimplicite (Shchepetkin 2015)

Configuration résolution ancien dt nouveau dt

BENGUELA (Penven et al) 25 km 6300s 7140s

OMAN (Vic et al) 2 km 160s 470s

Aucun contrôle sur les erreurs Obligation d’utiliser un schéma linéaire sur la partie explicite

Schémas C4 et « akima » (Van leer 1977)

V_ADV_C4, VADV_AKIMA

Comment choisir ???

Equation horizontal vertical

Moment 3D (UV_ADV)(+ eq W_ en NBQ)

UV_HADV_TVDUV_HADV_C2

UV_HADV_UP3 UV_HADV_C4

UV_HADV_UP5UV_HADV_C6

UV_HADV_WENO5

UV_VADV_TVDUV_VADV_C2

UV_VADV_SPLINES UV_VADV_WENO5

Traceurs

TS_HADV_UP3 TS_HADV_RSUP3

TS_HADV_C4TS_HADV_WENO5

TS_HADV_UP5TS_HADV_C6

TS_HADV_RSUP5

TS_VADV_TVDTS_VADV_C2

TS_VADV_SPLINESTS_VADV_AKIMA TS_VADV_WENO5

Moment 2D M2_HADV_UP3 M2_HADV_C2

+V_VADV_IMP +BIO_HADV_WENO + …

3920 combinaisons …

-> ne pas choisir choix robustes par défaut -> examiner les ordres sources d’erreur -> physique du pb :

positivité, échelles -> temps de calcul

En pratique

2 appels par par de temps -> réutilise (ou pas) les mêmes routines

Moment : 2 appels à rhs3d.F avec aussi:

- u_hadv_order5.h - uvhadv_tvd.h - uhdiff_order5.h

Traceurs : pre_step3d.F, step3d_t.Favec aussi :

- t3dadv_order5.h - compute_vert_tracer_fluxes.h

Mais aussi : schémas différents au prédicteur (moins cher)et au correcteurex : C6+WENO5, C4+UP3

# define PREDICTOR (traceurs) �

new /= 3 (moment)

t(nx,ny,nz,nt

u(nx,ny,nz,3)

Opérateurs visqueux et diffusion

Forme de l’opérateur visqueux (Wajsowic, 1993)

UV_VIS2 Tenseur visqueux (uv3dmix.F)

l’opérateur de viscosité est donné par

Assure :- conservation de la quantité de mouvement- conservation du moment angulaire- terme strictement dissipatif

UV_VIS4 Même logique appliquée 2 fois

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Fermeture de type Smagorinsky

UV_VIS_SMAGO, UV_VIS2

TS_DIF_SMAGO, UV_VIS2

Coefficient de viscosité turbulente

Coefficient de diffusion turbulente

Diffusion tournée (TS_MIX_ISO, TS_MIX_GEO)

TS_DIF2

TS_DIF4

TS_MIX_IMP : Method of stabilising correction

Diffusion tournée : illustration et limitations

Limitations : • Basé sur l’hypothèse faible pente• Diffusion non monotone (termes croisés)• Suppose une évolution basse fréquence des isoprènes

34.3 34.35 34.4 34.45 34.5 34.55 34.6Salinity [psu]

2

3

4

5

6

7

8

Tem

pera

ture

[Cel

sius]

SODA

ISONEUTRAL

ISOSIGMA

34.3 34.4 34.5 34.6 34.7Salinity [psu]

2

3

4

5

6

7

8

Tem

pera

ture

[Cel

sius

]

34.2 34.3 34.4 34.5 34.6Salinity [psu]

2

4

6

8

10

Tem

pera

ture

[Cel

sius

]

34.5 34.55 34.6 34.65 34.7 34.75 34.8Salinity [psu]

2

4

6

8

10

Tem

pera

ture

[Cel

sius

]

SODA

ESPCW WSPCW

WNPCWSPEW

SIG2

ISO2

En pratique

hmix_coef.F : coefficients

t3dmix.F: - t3dmix_ISO.F (isop ou geo) - t3dmix_S.F

t3dmix_tridiagonal.h

uv3dmix.F: - uv3dmix_S.F - uv3dmix_GP.F - uv3dmix4_S.F - uv3dmix4_GP.F

sur un pas d’Euler

step3d_t.F

step3d_uv2.F