A hybrid collocated/staggered version of the Fast Vector Penalty-Projection method for dilatable fluids

(1)

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A hybrid collocated/staggered version of the Fast Vector

Penalty-Projection method for dilatable fluids

Michel Belliard

To cite this version:

Michel Belliard. A hybrid collocated/staggered version of the Fast Vector Penalty-Projection method

for dilatable fluids. Comptes Rendus Mathématique, Elsevier Masson, 2014, 352 (9), pp.761-766.

�hal-02047696�

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Ve tor-Penalty Proje tion method for dilatable uids Mi helBelliard

a

,

a

Commissariatàl'

énergieatomique etauxénergiesalternatives

CEA,DEN/DANS/DM2S/STMF/LMEC

CEACadara he,Bât.

238

,F-

13108

StPaul-lez-Duran e

Re eived*****;a eptedafterrevision+++++

Presentedby£££££

Abstra t

WeproposeahybridversionoftheFastVPPmethodfordilatableuids: ollo atedvariables/staggeredproje tion.

The ne essary onditions for its ee tive appli ation are outlined. Numeri al results illustrate the signi ant

omputation- ostredu tiontorea hstationaryregimes.To ite this arti le:A.Name1,A.Name2,C.R. A ad.

S i.Paris,Ser.I340(2005).

Résumé

Une version hybride olo alisée/dé alée de la méthode Fast Ve tor-Penalty Proje tion pour les

uidesdilatables. OnproposeuneversionhybridedelaméthodeFast-VPPené oulementdilatable:variables

olo alisées etproje tiondé alée. Onpré iseles onditions né essaires à sonappli ation e a e.Des résultats

numériquesillustrentlegaineneortde al ulpourl'obtentionderégimesstationnaires. Pour iter etarti le:

A.Name1, A.Name2,C.R.A ad.S i.Paris,Ser.I340(2005).

Versionfrançaiseabrégée

La méthode Fast Ve tor-Penalty proje tion [1,2℄ permet de résoudre leséquations de Navier-Stokes

Eq. (1)-(4)pourunuidein ompressible.Un petit paramètredepénalité

0 < ǫ ≪ 1

ontrlela onser-vation de la masse. Son prin ipal intérêt réside en une résolution extrêmement rapide de l'étape de

proje tion.Anotre onnaissan e,lamiseen÷uvredesEqs(5)-(7)n'aétéfaitequ'envariablesdé alées.

On propose i i une versionvolumes nis hybride de la Fast-VPP pour des é oulements dilatables, f.

Eqs(9)-(11).Lesvariablesrestent olo alisées, f.Eqs(13)-(14),maislaproje tionestfaiteenvariables

(3)

tiale et temporelle on ernant l'étape deproje tion, f. Eq.(15),on analyselespropriétés dela matri e

orrespondante.Enparti uliersonnoyaun'estpasréduitauseulélémentnulet ellen'estpasfor ément

àdominan ediagonale

∀ǫ

.Ce i ades onséquen es surle solveurlinéaireutilisé pour larésolution. On indiqueles onditionsné essairesàl'appli atione a edelaméthodeILU(0)-BiCGStab,enparti ulier

ladétermination d'un

ǫ

opt

optimal, f. Eq. (17).Des résultats numériquesillustrent l'e a itéde ette appro he qui permet deréduire sensiblementl'eort de al ul duterme de pressionpourla simulation

derégimesstationnaires.Ce iesttoutparti ulièrementvraidansle asd'ungrandnombrededegrésde

liberté.

1. Introdu tion

We propose ahybrid version of the Fast Ve tor Penalty-Proje tion (VPP) method, see [1,2℄, in the

frameworkoftheFiniteVolume(FV) methodusing ollo atedvariablesonCartesiangridsfordilatable

uids.Themostattra tivefeatureofthefastVPP isits apabilitytoeasily solvetheproje tionstepin

omparisonwiththe lassi alChorin-Temam'smethod[4,7℄.TheNavier-StokesequationsforaNewtonian

uidin

Ω

,anopenof

R

d

,

d = 2, 3

,are:

∂

t

ρ + ∇ · (ρv) = 0

in

Ω × [0, T ]

(1)

∂

t

(ρv) + ∇ · (q ⊗ v) + ∇P − ∇ · τ (v) = g

in

Ω × [0, T ]

(2)

+Boundary onditionsfor

ρ

,

v

and

P

on

∂Ω × [0, T ]

(3)

+Initial onditionsfor

ρ

,

v

and

P

in

Ω

at

t = 0

(4)

with

ρ

thedensity,

P

thepressure,

v

theuidvelo ity,

q

= ρv

themassux,

τ (v) = µ((grad+grad

T

_)v−

2

3 divv I)

thevis ousstrainand

µ

thedynami vis osity.Atthesemi-dis retelevel,thein rementalform oftheVPPs hemeforin ompressibleuidsreadsas[1℄:

˜

v

n+1

− v

n

△t

+

1 ρ

DC(v

n

_{, ˜}

_v

n+1

_{) = −}

1 ρ

∇P

n

₊

g

n+1

ρ

(5)

ǫ(

v

ˆ

n+1

△t

+

1 ρ

DC(v

n

_{, ˆ}

_v

n+1

_{)) − ∇(∇ · ˆ}

_v

n+1

_{) = ∇(∇ · ˜}

_v

n+1

₎

(6)

1 ρ

φ

n+1

_{= −}

1 ǫ

∇ · v

n+1

(7)

where

△t

isthetimestep,

DC

thesemi-dis reteformofthediusion- onve tionoperatorfrom Eq.(2),

ˆ

v

n+1

:= v

n+1

_{− ˜}

_v

n+1

,

φ

n+1

_{= P}

n+1

_{− P}

n

and

0 < ǫ ≪ 1

the penalty parameter. For Eq. (5), the Boundary Conditions (BC)are taken from (3). BCs for

v

ˆ

are homogeneousDiri hlet onditionswhere the velo ity is pres ribed. Thanks to Eq. (7) it is possible to re over the pressure. This suggestsalso

homogeneousDiri hlet onditionsfor

∇ · v

where thepressureis pres ribed. Infa t, itis not usefulto re onstru tthepressureatea htimestep.We anuseEqs(6)and(7) toupdatethepressuregradient:

∇P

n+1

_{= ∇P}

n

_{+ ∇φ}

n+1

.

TheVPP methodsharessomelinkswiththeaugmentedLagrangian[6℄andthearti ial ompressibility

methods [3℄, see [1℄. Eq. (7)is similar to

1 ρ

φ

n+1

_{= −r∇ · v}

n+1

withanaugmentationparameter

r =

1 ǫ

. Angot et al.haveproventhe onvergen e ofthe velo ity divergen e

|∇ · v|

L

2

with theorder

O(ǫ)

fora given

△t

[1℄.

(4)

ǫ

△t

v

ˆ

n+1

_{− ∇(∇ · ˆ}

_v

n+1

_{) = ∇(∇ · ˜}

_v

n+1

_).

(8)

Unlike theVVP s heme, the Fast one doesnot provide asemi-dis rete form of Eq. (2) under the

on-straint (7). But its main interest is to be found in the easily solving of Eq. (8). With a small enough

valuefor thepenaltyparameter

ǫ

△t

, the righthand side of this linearsystemis formed using thesame matrix thanthe left hand side. If we denote by

W

the matrix asso iated to theoperator

∇(∇ · ∗)

, we have:

v

ˆ

n+1

₌

△t

ǫ

(I −

△t

ǫ

W )

−1

_{W ˜}

_v

n+1

and

v

ˆ

n+1

_{≈ −W}

−1

_{W ˜}

_v

n+1

for

0 <

ǫ

△t

≪ 1

. Using aniterative solverforthislinearsystem,we antakeadvantageofthisfeaturebyanappropriatedpre onditioningas

theILUone.Letsnoti ethat the

W

matrixisanon-symmetri oneandwehaveto hoosetheiterative solverunder this onstraint.Here we onsidertheBiCGStabiterativemethod.

Hereafter, we detail ourFV hybrid ollo ated/staggered versionfor dilatableuids on Cartesian grids

and give some numeri al illustrations showing the apa ity of our approa h to ee tively redu e the

omputational osttorea hstationaryregimes.

2. A hybrid ollo ated/staggered Fast VVP

Atthesemi-dis retelevel, theproposedin remental formof theFastVPP s hemefordilatableuids

(orvariable-densityows)readsas:

˜

q

n+1

− q

n

△t

+ DC(ρ

n+1

_{, v}

n

_{, ˜}

_v

n+1

_{) = −∇P}

n

_{+ g}

n+1

(9)

ǫ

△t

q

ˆ

n+1

_{− ∇(∇ · ˆ}

_q

n+1

_{) = ∇(∇ · ˜}

_q

n+1

₊

ρ

n+1

− ρ

n

△t

)

(10)

∇φ

n+1

= −

q

ˆ

n+1

△t

(11) where

q

n+1

_{:= ρ}

n+1

_v

n+1

,

q

˜

n+1

_{:= ρ}

n+1

_v

_˜

n+1

and

q

ˆ

n+1

_{:= ρ}

n+1

_v

_ˆ

n+1

_{= q}

n+1

_{− ˜}

_q

n+1

.Let's remarkthat

thissystemdiersfromtheonegivenin[1℄forageneralizationtovariable-densityows:

ρ

n+1 ˜

v

n+1

−v

n

△t

+

. . . = −∇P

n

_{+ g}

n+1

and

ǫ

△t

q

ˆ

n+1

− ∇(∇ · ˆ

v

n+1

) = ∇(∇ · ˜

v

n+1

)

.Theevolutionofthedensityissupposed givenbyanextrabalan eequation;forinstan e,theenergybalan eequationequippedwiththeequation

ofstate

ρ(e)

.ConfrontingEq.(10)andEq.(11),wegetanequationinvolving

φ

and

q

ˆ

,thatissimilarto Eq.(7):

φ

n+1

= −

1 ǫ

[∇ · q

n+1

₊

ρ

n+1

− ρ

n

△t

].

(12)

It an be viewed as a parti ular form of arti ial ompressibility applied to the onstraint

∇ · q =

−

ρ

n+1

_△t

−ρ

n

.

We onsideraFVspa edis retizationwith ollo atedvariables(density,velo ity,massuxandpressure)

atthe ell entersofaCartesiangrid.Hereafter,wedenoteby

K

a elland

σ

a ellfa e.Notation

∗

K

or

∗

σ

willstandsfora ellor afa e variable,respe tively.TheVPP method wassu essfullytestedwitha staggered-variables heme onstru tured[1℄orunstru tured[2℄meshes. Forstaggered-variables hemes,

theFV approximationofthe

∇ · q

operatorislo atedat the ell enters-denoted

∇

K

· q

σ

-and allthe neededdataareavailable:

q

σ

· n

σ

with

n

σ

thefa enormal.TheFVapproximationofthe

∇(∗)

operator -denoted

∇

σ

(∗

K

)

- is lo atedat the fa e enters. It gives the orre tion eld forwhi h onlythe normal omponents

q

ˆ

σ

· n

σ

areusefulandtheBCswelldened.

(5)

UsingtheVPPmethodwith ollo atedvariablesisnotsoeasy.For ollo ated-variables hemes,the

∇ · ∗

operatorisapproximatedatthefa e enters

∇

σ

·∗

K

andthe

∇(∗)

oneatthe ell enters

∇

K

(∗

σ

)

.Thereis noproblemtodenedthelatteroperator

∇

K

(∗

σ

)

,butitisnotthe asefortheformerone

∇

σ

·∗

K

.Indeed, wehavenospe i ationfortheboundary onditions on erningtheneededtangentialvelo ities:theterm

∂qτ

∂τ

with

τ

theboundarytangent.Here,weover omethisissueusingahybrid ollo ated/staggeredsolver forwhi htheproje tionstepis doneonthe ellfa es.Infa t,itwasalreadythe aseintheFVs heme

presented byFaure et al. [5℄.The hybridmethod that weproposeapplies a ollo ated-variables heme

forthepredi tedvelo itystepandastaggered-variables hemefortheproje tionstep.

The ell- entervelo ities

v

˜

K

verifythemomentumbalan e equationusing onve tionuxes

q

σ

= ρ

σ

v

σ

atthefa e enters:

|K|

ρ

n+1

K

˜

v

n+1

K

− ρ

n

K

v

n

K

△t

+

X

σ∈ǫ(K)

|σ|˜

v

_σ

n+1

q

n

_σ

· n

σ

− D

K

(˜

v

n+1

K

, ˜

v

n+1

L

) = −∇

K

P

K

n

+ |K| g

K

n+1

(13)

with

ǫ(K)

thefa es of the ell

K

,

| ∗ |

themeasure of

K

or

σ

,

D

K

(∗)

aspa e-time dis retizationof the diusion term (with ells

L

in the vi inity of

K

) and

v

˜

σ

givenby an appropriated interpolation. The normal omponentsofthe onve tionuxes

q

σ

· n

σ

verifythemassbalan eequation(andnotthe

ρ

K

v

K

themselves):

|K|

ρ

n+1

K

− ρ

n

K

△t

+

X

σ∈ǫ(K)

|σ|q

n+1

σ

· n

σ

= 0.

(14)

Hen e,the orre tions

q

ˆ

liveon

σ

andweonlyneedthenormal omponents

q

ˆ

σ

· n

σ

givenbythenormal dotprodu t of Eq. (10). Lets

˜

q

n+1

σ

= ρ

n+1

σ

v

˜

n+1

σ

be amass ux predi tioninvolvingappropriated fa e interpolationsof

ρ

K

and

v

˜

K

on

σ

.Then,we onsiderthefollowingFVapproximationofthenormaldot produ tofEq.(10):

ǫ

△t

q

ˆ

n+1

σ

· n

σ

− ∇

σ

(

∇

K

· ˆ

q

n+1

σ

|K|

) · n

σ

= ∇

σ

(

∇

K

· ˜

q

n+1

σ

|K|

+

ρ

n+1

_K

− ρ

n

K

△t

) · n

σ

.

(15)

Inthisequation,the elldivergen eandthenormalfa egradientaredenedbylinearinterpolations(but

higher-orderinterpolations anbeused):

∇

K

· ∗

σ

=

X

σ∈ǫ(K)

|σ|∗

σ

· n

σ

and

∇

σ

(∗

K

) · n

σ

=

∗

K

− ∗

L

h

σ

.Here

K

and

L

arethe ellssharingthefa e

σ

andsu hthat

n

σ

isdire tedfrom

K

to

L

and

h

σ

isthenormal distan e between the

K

and

L

ell enters. As a onsequen e, Eq.(15) givesonly a ess to the normal omponentsof

q

ˆ

σ

.As alreadymentioned,the BCsfor this quantityarewelldened.Whenthe normal inowmassuxisimposed(

˜

q

σ

· n

σ

= q

σBC

· n

σ

),wehave

ˆ

q

σ

· n

σ

= 0

.Andwhentheoutowpressureis imposed (

P

σ

= P

σBC

;

φ

σ

= 0

),weget

∇ · (q +

∂ρ

∂t

)

σ

= 0

fromEq.(12). On e that

q

ˆ

n+1

σ

· n

σ

is obtainedfrom Eq. (15),thenormal omponentsof the ell-fa emassux are updated:

q

n+1

σ

·n

σ

= ˜

q

n+1

σ

·n

σ

+ ˆ

q

n+1

σ

·n

σ

.Similarly,Eq.(11)providestheupdateofthenormal omponent ofthe ell-fa epressuregradient

∇

σ

(P

n+1

K

) · n

σ

= ∇

σ

(P

K

n

) · n

σ

−

q

ˆ

n+1

σ

△t

· n

σ

.But,forthe ollo ated ell- entered FVs hemeused withthemomentum balan e equation, weneedto update the ellquantities.

Hen e,wedenethefollowinglinearinterpolationofthe omponents:

q

ˆ

n+1

K

·ξ =

ˆ

q

n+1

σ+

·n

σ+

−ˆ

q

n+1

σ−

·n

σ−

2

where

ξ

istheunitaryve torin the onsidered omponentdire tionand

σ

+

(resp.

σ

−

)a ellfa e of

K

su has

n

σ

+

= ξ

(resp.

n

σ

₋

= −ξ

).Then,weupdatethe ellmassux by

q

n+1

K

= ˜

q

n+1

K

+ ˆ

q

n+1

K

,the ellvelo ity by

v

n+1

K

=

q

n+1

_K

(6)

∇

K

P

_K

n+1

= ∇

K

P

K

n

−

ˆ

q

n+1

_K

△t

.

(16)

Thepropertiesofthenon-symmetri matrix

[

ǫ

△t

− ∇

σ

(

∇K·

|K|

)]

,denotedby

ǫ

△t

I − W

,resultfromthelinear approximationsof the operators

∇

K

·

and

∇

σ

. As its Kernel is notredu ed to thesingle

{0}

,spurious wavesmayo urwithoutstabilizationpro ess(notdis ussedhere).Moreover,as

X

i6=j

|a

(i,j)

| = 10/h

2

and

|a

(i,i)

| = 2/h

2 + ǫ/△t

, the matrix

ǫ

△t

I − W

is a non-diagonal dominant one unless

ǫ

△t

is big enough. Here,

i

and

j

standfortherowand olumnindexes(twotabular-indextriplets) and

a

(i,j)

forthematrix oe ients.We andetermineanoptimalvalueforthepenaltyparameter

ǫ

△t

inordertogeta diagonal-dominantmatrix

ǫ

△t

I − W

.Inthelimit

ǫ

△t

≈ 0

,theaboveequationsshowthat thesumof theabsolute values of the extra-diagonalterms is

r

a

= 5

times bigger than the diagonal term itself. Hen e we an determine

δa

(i,i)

=

ǫopt

△t

in su h away that this ratiowould be theunity:

|a

(i,i)

+

ǫ

opt

△t

| =

X

i6=j

|a

(i,j)

| =

r

a

|a

(i,i)

|

.Thisrelationleadsto

ǫ

opt

= (r

a

− 1) max|a

(i,j)

| △t

thatis:

ǫ

opt

≈ 8 max{

1 h

2 x

,

1 h

2 y

,

1 h

2 z

} △t.

(17)

Obviously

ǫ

opt

nomoreveries

0 < ǫ ≪ 1

.Indeed,thisapproa hisrelevantofthe lassoftheaugmented Lagrangianmethods [6℄withanaugmentedLagrangianterm

r∇(∂

t

ρ + ∇ · q)

and

0 < r =

1 ǫopt

.

3. An example ofnumeri al appli ation

Ourpurposeisnottowidely he kourproposedhybrids hemebutonlytoprovideaveryrstinsight

initsappli ationtodilatableuids omputedwith ollo atedvariablesonCartesiangrids.Ourtest ase

onsistsin a2Dstationarydilatablevortexow( omputedin 3D).The3D omputationaldomain

Ω

is a ubeoflength one, enteredin

(−0.5, −0.5, 0.5)

t

. Thestationary regimeis foundby atimemar hing

algorithmin

[0, T ]

:

∂

t

w + ∇ · (wv) = 0

in

Ω × [0, T ]

(18)

∂

t

ρv + ∇ · (ρv ⊗ v) − ∇ · τ (v) + ∇P = 0

in

Ω × [0, T ]

(19)

ρ(x, y, z) = (ρ

1 − ρ

0 )w(x, y, z) + ρ

0

(EOS;

ρ

0 = 1

and

ρ

1 = 2

). (20)

The stationary analyti al solutionis:

w(x, y, z) =

1

2 (x

2 + y

2 )

,

v

(x, y, z) = (−y, x, 0)

T

and

P (x, y, z) =

1

8 (x

2 + y

2 )

2 +

1

2 ρ

0 (x

2 + y

2 ) + P

0

with forinstan e

P

0 = 0

. This solutionis apolynomialoneof degree twoin

w

and

ρ

, of degreeonein

v

and of degreefour in

P

. And wehave

∇ · v = 0

.At theinow,we pres ribeDiri hletBCfor

w

and

v

and homogeneousNeumannonefor

P

.Attheoutow,wepres ribe Diri hletBCfor

P

andNeumannonesfor

w

and

v

.Thetransposedpartofthestresstensor(

µ = 10

−3

)

isnot onsideredhere.

ThesystemEqs(18)-(20) issolvedin theresteps.Therst oneis devoted tothe omputation of

w

n+1

K

,

ρ

n+1

_K

andtheinterpolation

ρ

n+1

σ

.Withthese ondone,weobtainapredi tedvelo ity

v

˜

n+1

K

andthe inter-polation

v

˜

n+1

σ

.Finally,the orre tionmassux

q

ˆ

n+1

σ

is omputedaswellasthe orre tedmassux

q

n+1

σ

, theinterpolated orre ted-velo ity

v

n+1

K

andtheinterpolated orre tedpressuregradient

∇

K

P

n+1

K

.Given aninitialguess

∇

K

P

0 K

,thepressuregradientisstraightforwardlyobtainedfromEq.(16).(In the aseof theChorin-Temams heme,

∇

K

P

K

:=

X

σ∈ǫ(K)

(7)

thedire tlyavailablepressure

P

K

.)

AsawholetheFastVPPs hemeprodu esquantitativelysimilarresultstothoseoftheChorin-Temam

s heme.Table3givesa ompilationoftheresultsobtainedwiththelinearinterpolationsfortheoperators

∇

K

·

and

∇

σ

,theupwinds hemeforthe onve tionof

v

andof

w

.Thea ura yandthespa e onvergen e order

O(h

1 ₎

oftheerrorsinthe

L

2

and

L

∞

normsarekeptforthe

w

andthe

v

variables.Thislatterpoint is alsotruefor the

L

2

norm of thepressure-gradienterror. Butthepressureboundarylayersare larger

than those obtained with the Chorin-Temam s heme. It limitsthe pressure-gradienterror onvergen e

orderin

L

∞

norm.Letsnoti ethat substitutingthelinearinterpolationformulaeto ompute

q

ˆ

n+1

K

bya quadrati formulaeisawaytoin reasethesmoothnessofthepressure-gradienterrordistributionandto

redu ethemagnitudeofthepressureboundarylayers.

Grid

|e

w

|

L

2 |e

w

|

∞

|e

v

|

L

2 |e

v

|

∞

|e

∇P

|

L

2 |e

∇P

|

∞

S heme (ratio) (ratio) (ratio) (ratio) (ratio) (ratio)

10x10x5 3.88

10 −2

7.58

10 −2

2.64

10 −2

4.37

10 −2

1.63

10 −1

5.73

10 −1

FastVPP 20x20x5 2.18

10 −2

4.89

10 −2

1.39

10 −2

2.81

10 −2

8.12

10 −2

3.52

10 −1

FastVPP (1.78) (1.55) (1.90) (1.56) (2.01) (1.63)

ˆ

q

n+1

K

quadrati interpolation2.18

10 −2

5.30

10 −2

1.39

10 −2

3.12

10 −2

3.73

10 −2

1.96

10 −1

40x40x5 1.19

10 −2

2.93

10 −2

7.33

10 −3

1.38

10 −2

3.44

10 −2

3.32

10 −1

FastVPP (1.83) (1.67) (1.90) (2.04) (2.36) (1.06) 10x10x5 3.88

10 −2

7.61

10 −2

2.56

10 −2

4.29

10 −2

1.42

10 −1

6.42

10 −1

Chorin-Temam 20x20x5 2.19

10 −2

4.91

10 −2

1.38

10 −2

2.80

10 −2

6.40

10 −2

3.46

10 −1

Chorin-Temam (1.77) (1.55) (1.86) (1.53) (2.22) (1.86) 40x40x5 1.14

10 −2

2.80

10 −2

7.21

10 −3

1.53

10 −2

2.89

10 −2

1.94

10 −1

Chorin-Temam (1.92) (1.75) (1.91) (1.83) (2.21) (1.78) Table1

Absolute errors and spa e onvergen e orders in

L

2

and

L

∞

norms for the Fast VPP proje tion s heme using linear

interpolationsfortheoperators

∇

K

·

and

∇

σ

.ComparisonwiththeChorin-Temamproje tions heme.Upwinds hemefor the onve tionof

v

and of

w

.Linear interpolationfor

q

ˆ

n+1

K

.CFL =0.25.Grids 10x10x5and20x20x5:800

∆t

;40x40x5: 1500

∆t

.

As laimedin[1℄forin ompressibleows,

ǫ

providesa ontrolonthe

L

2

normofthevelo itydivergen e.

The de reasingof the velo ity divergen e error is ontrolled by a

O(ǫ∆t)

term. Here,we an expe t a similarbehaviorforthemassbalan eerror

|∂

t

ρ + ∇ · q|

L

2

. Usingaxedtime step,Fig.1(a)illustrates thispointforthe

L

∞

norm.On e

ǫ

issmallenough,thiserrorisdrasti allyredu edinalinearway.The numberof unpre onditioned BiCGStab iterationsto solvethe linearsystem is quitelarge: e.g.at time

stepnumbertwo,about75 whatever

ǫ

is. TheChorin-Temam algorithmneeds aboutthesamenumber ofConjugateGradient(CG)iterationsforthistimestep.Moreover,theBiCGStab methodrequirestwo

matrixve torprodu tsbyiterationinstead ofonlyonefor theCG.Hen e,pre onditioningis required.

Usingthemaximalpivotalgorithmto omputetheILU(0)pre onditioningmatrix,wegetsmallvaluesfor

somediagonalentries- around

10 −12

-leadingto the omputationfailure. Choosing theoptimalvalues

ǫ

opt

, we re over adiagonal-dominantmatrix. For instan e, with the grid 20x20x5and

△t = 5.6 10

−3

se ond, a numeri al appli ation of Eq. (17) gives

ǫ

opt

≈ 17.9

and

r =

1 ǫopt

≈ 6 10

−2

. In this ase,

thesmallestdiagonal entryof theILU(0) matrixis about

10 −1

(8)

BiCGStab).However,themassbalan eresidual, ontrolledbyEq.(12),isrelativelyhigh(

10 −3

to

10 −4

)

at thebeginningof thetransientbut qui kly de reasesduring thetransient(

φ

n+1

is nul forthesteady

state), f.Fig.1(b).

ThenumberofBiCGStabiterationsareamazinglyredu edwiththeILU(0)pre onditioning.Indeed,the

transient-averagednumberof ILU(0)-BiCGStabisindependentofthegridresolution: about3whatever

thegrid size;insteadof27.7(10x10x5) to191.5 (80x80x5)BiCGStab iterationsfortheChorin/Temam

s heme. Con erningthetransient CPUtime, thespeed-upof theoverall omputation is betweenabout

1.6( oarsestgrid)and2.7(nestgrid)in omparisonwiththeChorin/Temam omputation.

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 epsilon

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01 mass balance error

y = 0.0055928 * x^0.98586

(a) Mass balan e error in

L

∞

norm versus

ǫ

. Time step numbertwo.Upwinds hemeforthe onve tionof

w

.

0

500 1000

1500

2000

Time iteration

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01 Mass balance error in Linf norm

(b) ILU(0)-pre onditioned BiCGStab method: mass

bal-an e error in

L

∞

norm versus the iteration number.

ǫ

=25.28.QUICKs hemefor onve tionof

w

.

Figure 1.Fast VPP mass balan e errors. Linear interpolations for the operators

∇

K

·

and

∇

σ

.Upwind s heme for the onve tionof

v

.Grid20x20x5.CFL=0.25.

∆t

≈ 8 10

−3

se ond.

Referen es

[1℄ Ph. Angot, J.-P. Caltagirone, and P. Fabrie. Ve tor Penalty-Proje tion Methods for the Solution of Unsteady

In ompressibleFlows. In FiniteVolumes forComplex Appli ations V,pages 169176, Aussois,Fran e, June 08-13

2008.Wiley.

[2℄ Ph.Angot,J.-P.Caltagirone,andP.Fabrie. ASpe ta ularVe torPenalty-Proje tionMethodsforDar yans

Navier-StokesProblems. InFiniteVolumesforComplexAppli ationsVI,Prague,Cze hRepubli ,July01-232011.Wiley.

[3℄ A.J.Chorin.Anumeri almethodforsolvingin ompressiblevis ousowproblems.JournalofComputationalPhysi s,

2:1226,1967.

[4℄ A.J.Chorin.Numeri alsolutionofthenavier-stokesequations. Mathemati sofComputation,22:745762,1968.

[5℄ S.Faure,J.Laminie,andR.Temam. Colo ated FiniteVolumeS hemesforFluidFlows. Commun.Comput.Phys.,

4(1):125,2008.

[6℄ M.Fortinand R.Glowinski. AugmentedLagrangianMethods:Appli ationstotheNumeri alSolutionof

Boundary-ValueProblems,volume15ofStudiesinMathemati sandItsAppli ations. North-Holland,Amsterdam,1983.

[7℄ R.Temam. Surl'approximationdelasolutiondeséquationsdeNavier-Stokesparlaméthodedespasfra tionnaires.