finite volume method in curvilinear coordinates for hyperbolic conservation laws

Sangam 7 Thispaperdealswiththedesignofnitevolumeapproximationofhyperb ation laws in ordinates.. Inthis paperwe derive anew nite volumemethodfor hyperb ation laws in ordinates.. Asa Carte

E N.Crouseilles,H.Guillard,B.Nkonga,andE ker,Editors

∗

A Bonnement

1

, T Fajraoui

2

, H Guillard

3

, M Martin

4

, A Mouton

5

, B

Nkonga

6

and A Sangam

7

Thispaperdealswiththedesignofnitevolumeapproximationofhyperb ation

laws in ordinates h ordinatesare terednaturally inmanyproblemsas for

in the analysis of a large number of models from t fusion in

tokamaks Inthis paperwe derive anew nite volumemethodfor hyperb ation laws in

ordinates Themethodis rst ed inageneral settingand thenis illustrated in

2Dpolar ordinates experimentsshowitsadvantageswithresp totheuseofCartesian

ordinates

1 Intr

ationlaws h hemesare whenoneisinterestedin thepropertiesofthe

ph model under in h ordinatessystemplay animportantrole Theph models

ofinterestarefor those harged motioninsolarwindsintheframeofastroph

plasmas[4℄orthetransportof harged inatokamak,a FusionConnement

totheignitionof trolledthermon fusion onearth[4,8,10℄

More ,in magnetizedplasma,therearetwo behavioursof alongand

mag-eldlines Thisleadstohighly owsoftheplasma Asa Cartesian ordinates

donot anappropriatesystemto etheph thattakes intheplasma Instead,other

systemsof ordinatesarepreferred,asfor eldaligned ordinatessystems[1,2℄,Boozer ordinates

or Hamada ordinates[6℄ Theeld governingequations writtenin these generalized systemsare

generally not in ation laws form: spatially varying ts multiply the dierential

termsandadditional termsappearin theequations Thereforethedesignofanitevolumemethod is

notasstraighforwardasitisin Cartesian ordinatesand additionallyimportant ationproperties

belostb the Anotherrelevantquestionthat arisesin this text therepresentation

∗

A.BonnementwassupportedinpartbytheConseilRégionalPr e-Alpes-Cted'Azur

1

INRIASophia-Antipolis&Université Sophia-Antipolis,UMRCNRS6621(F e-mail:audrey.bonnemen

2

UniversitéV FRCNRS2956(F e-mail:tarek.fajraouiuniv-v

3

INRIASophia-Antipolis&Université Sophia-Antipolis,UMRCNRS6621(F e-mail:Herve.Guillardinria.fr

4

Université Sophia-Antipolis,UMRCNRS6621&INRIASophia-Antipolis(F e-mail:

5

UniversitéToulouse3,UMRCNRS5219(F e-mail:alexandre.moutonmath.univ-toulouse.fr

6

Université Sophia-Antipolis,UMRCNRS6621&INRIASophia-Antipolis(F e-mail:b

7

Université Sophia-Antipolis,UMRCNRS6621&INRIASophia-Antipolis(F e-mail:afein

Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/2011019

ofv andthe ofthebasisin hthesev areexpressed in ordinates,the

basisarespatiallydependent For thepro ofav inthelo basisofthe onding

ordinatessystemintro terms fromthevariationsoflo basiswithresp tothevariables

ofthe hosen ordinates,and the ationlawsformoftheequationis thereforelost Froma

n pointofview,ndinganappropriateapproximationofthiskindoftermsthatkeepsthe ation

propertiesof the system ofequations remains a hallenge, for this purpose it is useful to think about

termsinshallowwatersystems,ortoCoriolis termingeoph equations[7℄

gen-eral ordinates, withoutany preliminarpro when dealingwith v equations Averaged

termsina tmannerindependentlyofthe systemused

Thispaperis organizedasfollows In 2,prerequisiteson ordinatesare Finite

volumemethodsinthese ordinatesaredesignedin 3 testsusingtwo-dimensional

ordinatesasexamplearethen in 4inordertoillustrate our h Finally,

isgivenin 5

Letus aph modeldenedonaph domainΩ(x) ⊂ R 3

,where hpointM (x)ofΩ(x)is

lo b itsCartesian ordinatesx = t (x 1

, x 2 , x 3 ) Suppose nowtheph model under

beeasily ed inanother ordinatessystems, sothat theph model belookedthroughthe

domain Ω(ξ),where ξ = t (ξ 1

, ξ 2 , ξ 3 ) Thedomain Ω(ξ)will bereferredto asthe domain, and the onding ordinates systemξas ordinates Obviously, thereexists anone-to-one map

φ : ξ 7→ x, h isassumed to be at leastaC 1

-dieomorphism, h means that J the determinant ofthe matrixMJ ofφdenedb

M J =











∂x 1

∂ξ 1 ∂x 1

∂ξ 2 ∂x 1

∂ξ 3

∂x 2

∂ξ 1

∂x 2

∂ξ 2

∂x 2

∂ξ 3

∂x 3

∂ξ 1

∂x 3

∂ξ 2

∂x 3

∂ξ 3











is positive Tointro the expressions ofthe gradientand the div operators, ∇, ∇ ·, with resp to ordinatesξthatwillbeusedinthispaper,itisusefultodenethelo variantbasis

ek asso tothetransformationφgivenb

ek = ∂x

∂ξ k = ∂x

1

∂ξ k i + ∂x

2

∂ξ k j + ∂x

3

∂ξ k k ,

wherek = 1, 2, 3,andi,j andkarev ofthe basis ondingto theCartesian ordinates system The travariantbasise k

asso toe k isprovidedthroughtherelations

e k · ej = δ j k ,

whereδ k

j isthe kertensor

With thesequantities,thegradientofthev eldV (ξ)isgivenb

∇ V = ∂V

∂ξ k ⊗ e k =  ∂V

i

∂ξ k + V m Γ i

mk



ei ⊗ e k

(The Einstein summation vention is assumed through this paper.) Here Γ i

mk are the Christoel symbols givenb

∂em

∂ξ k = Γ i

mk ei ,

andrepresentthepro ontoei ofthe hangeofthev em toξ k

Thediv ofthev eldV (ξ)isdenedasthe ofthegradient

∇ · V = ∂V

∂ξ k · e k =  ∂V

i

∂ξ k + V m Γ i

mk



ei · e k = ∂V

k

∂ξ k + V k Γ i

ki

Byusingtheidentity

1 J

∂J

∂ξ k = Γ i

ki,onegetsthe expression

∇ · V = 1

J

∂(JV · e k )

∂ξ k

ConsideringatensoreldT,itsgradientisgivenb

∇ T = ∂T

∂ξ k ⊗ e k

Theabo erelation beexpandedasfollows

∇ T =  ∂T

ij

∂ξ k + T mj Γ i mk + T im Γ j mk



Thediv ofthetensoreldT isgivenb

∇ · T = ∂T

∂ξ k · e k

Usingrelation(2.1)thisleadsto

∇ · T = 1

J

∂

∂ξ k J T · e k

Weare nowready to designanite volume methodsin ordinatesforhyperb ation

laws

Letus ageneralhyperb ationlawequationswritten ina ordinatefreemanneras

∂W

∂t + ∇ · F (W ) = 0 ,

whereW isthestatevariable andF (W ) isitsux

Letusalso a transformationφ : ξ 7→ x,whose determinantof isJ Usingthe resultsoftheprevious andnotingthat∂tJ = 0,it beseenthatinthis ordinatessystem,theabo e equationb

∂J W

∂ξ k



J F (W ) · e k



= 0

Inordertoseethe touroftheproblem,wewillstudyseparatelythe ofnitevolumemethod

W

ux F (W ) isav Ina step,wewill the whereW isav whileitsuxF (W )is

atensor

∂(J S)

∂ξ k



J V · e k



T examplesfor aretheequationsof tinuityandenergyinuid

A tothenitevolumephilosophy,the equationsaresimplyobtainedb integrating(3.2)on

volumes(Ωi) i∈ N Then integrating equation (3.2)o era Ωi and dividingthe resultb the volume |Ωi|, onegets

1

|Ωi|

Z

Ω i

∂(J S)

|Ωi|

Z

Ω i

∂

∂ξ k



J V · e k



dΩ = 0 ,

h berewrittenas

∂

∂t

 1

|Ωi|

Z

Ω i

J S dΩ



|Ωi|

Z

Ω i

∂

∂ξ k



J V · e k



dΩ = 0

Intro thea erageS i = 1

|Ω i | R

Ω i J S dΩyields,

∂

∂t

 Si



|Ωi|

Z

Ω i

∂

∂ξ k



J V · e k



dΩ = 0

Theuxtermisalsoimmediately b thediv theorem,onehas

Z

Ω i

∂

∂ξ k



J V · e k



dΩ = Z

∂Ω i

where ∂Ωi istheboundaryofΩi,nistheoutwardpointingunit v normaltothe ∂Ωi,and dσ(Ω)

theLebesguemeasureonthis Therighthandsideof(3.3)isimmediately assoonasonehas

n uxes[3,5,9℄ For this itisreadilyseenthat thereisno betweenthe of

nitevolumemethodin ordinatessystemandinaCartesianone

This dealswithW = V av andF (W ) = T atensor, thenthehyperb equationturns into

∂(J V )

∂ξ k



J T · e k



Momentum equationinuid is hakindofequations

Byusingthesamepro asin onegetsthefollowing heme,

∂

∂t

 1

|Ωi|

Z

Ω i

J V dΩ



|Ωi|

Z

Ω i

∂

∂ξ k



J T · e k



and atrst this seemssimilar tothe one However, V isav ithastobestored

equation(3.4)b thebasisv e k

(resp e k)andthentoobtain equationsforthe variant onents

ofthe v eldV k

(resp travariant onentsV k) Then these equationsare using the resultsof 3.1 In thesequel, wewill designate this method asthe pro tegrationmethod

This hhasoneimportant b thebasisv are spatiallydependent, theydonot

utewiththedierentialoperatorsandtherefore termsappearintheequations(seeequation(4.21)

for Theapproximationofthesetermsis andmoreoveritdependsonthesp

systemused

We therefore advo the use of the following pro that we will be the integration-pro

method

Firstwedene ana eragebasisinthe Ωi by:

ei,k = 1

|Ωi|

Z

Ω i

J ek dΩ ,

sothatoneobtains,assumingthatVi,k is tin a

1

|Ωi|

Z

Ω i

J V dΩ = Vi,k ei,k

Here, ei,k is the kth a eragev in the Ωi withresp to the hosen ordinates, and b denitionVi,krepresentsthea eragevalueofV alongthekthv tothe onding

ordinates The nitevolumeapproximationisthendened b

∂

∂t V i,k + e

i,k

|Ωi| . Z

Ω i

∂

∂ξ k



J T · e k



where(e i,k )k isthe travariantbasisasso to(ei,k)k

This pro is quitesimpleand itallowsforageneral(and ofthe terms In

thenext wedetailitinthe of2Dpolar ordinates

The ofnite volumemethodproposedin 3isillustratedin 2Dpolar ordinates

4.1 2D polar ordinates and nite volume method

Letus 2Dpolar ordinatesdenoted b (r, θ) ∈ (0, +∞[×[0, 2π)relatedtoCartesian ordinates

(x, y)b (x, y) = φ(r, θ)as follows,



x = r cos θ ,

y = r sin θ

We the usual orthonormal basis (ex , e y) of R 2, and then the variant basis asso with the ordinates(r, θ)isgivenb

e r =

 cos θ sin θ

 , e θ =



−r sin θ

r cos θ



The determinant asso to the transformation φ is J = r while the travariant basis with resp to

(er, eθ)isgivenb

e r =

 cos θ sin θ

 , e θ = 1

r



− sin θ cos θ



Working with the variant v e θ and the travariant v e θ

leads to a r, it is then appropriateto theirasso unit v

˜

eθ = 1

r eθ , e ˜ θ = r e θ

Then,for av V bewritten asV = V r e r + V θ e ˜ θ

Equippedwiththesenotations,we writedowntheexpressionsofgradientanddiv operators The

gradientofa S in polar ordinateswrites

∇ S = ∂rS e r + 1

r ∂θS ˜ e

θ

Letuswritethev V asV = V r er + V θ eθ ˜,itsdiv isgivenb

∇ · V = 1

r ∂r(r V r ) + 1

r ∂θ(r V θ ) = ∂rV r + 1

r V

r + ∂θV θ

T = T r,r er ⊗ er + T r,θ er ⊗ ˜ eθ + T θ,r eθ ˜ ⊗ er + T θ,θ eθ ˜ ⊗ ˜ eθ

Thediv ofT isgivenb thefollowingformula,

∇ · T = 1

r ∂ r(rT · e r ) + 1

r ∂ θ(rT · ˜ e θ )

= 1

r ∂ r(r T r,r er + r T r,θ eθ) + ˜ 1

r ∂ θ(r T θ,r er + r T θ,θ eθ) ˜

Now,letV = V r e r + V θ e ˜ θbeav htemporalevolutionisgovernedb ,

∂(J V )

∂ξ k



J T · e k



whereT isatensor

Applyingthepro developedin theprevious onegetsthefollowing heme,

∂Vi,r

∂t e i,r + ∂Vi,θ

∂t e i,θ + 1

|Ωi|

Z

Ω i

∂

∂ξ k



J T · e k



whereVi,r and Vi,θ area eragevaluesofV r andV θ,resp elyin the Ωi,whileei,r andei,θ area erage

v inthe Ωi,

ei,r = 1

|Ωi|

Z rer dΩ , ei,θ = 1

|Ωi|

Z

r eθ ˜ dΩ

1 2 3 4

A

B

C

D

θ A θB

Fig1: A inpolar ordinates

For y,thetensorT isassumedtobe i.e T r,θ = T θ,r

Wealso hoseatensorialmeshso

that the Ωi belo b thesegmentspro [rD , r A] × [θA , θ B ](seeFigure1) Then thea erage

v ei,r andei,θ beexpressedas

e i,r = 1

θ B − θA ( ˜ e θ A − ˜ e θ B ) , ei,θ = 1

θB − θA (er B − er A ) ,

where

er A = er D =

 cos θA sin θA

 , eθ ˜ A = ˜ eθ D =



− sin θA cos θA

 ,

and

e r B = er C =

 cos θB sin θB

 , e ˜ θ B = ˜ e θ C =



− sin θB cos θB



Finally,forany f = f (r, θ),weusethefollowingapproximations:

ˆ f| rD ≈ f (rD, θ) , f| ˆ rA ≈ f (rA, θ) , ∀ θ ∈ [θA, θB] , ˆ

f| θA ≈ f (r, θA) , f| ˆ θB ≈ f (r, θB) , ∀ r ∈ [rD, rA]

Thenequation(4.8)b

|Ωi| (∂tVi,r ei,r + ∂tVi,θ ei,θ) +  rA T ˆ r,r

| rA − rD T ˆ | r,r

rD

 (eθ A − eθ B ) +  rA T ˆ r,θ

| rA − rD T ˆ | r,θ

rD

 

er B − er A



+ (rD − rA)  ˆ T r,θ

| er B + ˆ T | θ,θ eθ B − ˆ T | r,θ er A − ˆ T | θ,θ eθ A



= 0

(4.9)

Itisinterestingtoexpandequation(4.9)onthepairoforthogonalv e i,r ande i,θ, hyields

|Ωi| ∂tVi,r + (θB − θA) (rA T ˆ | r,r

rA − rD T ˆ | r,r

rD ) + θB − θA

2

sin(θB − θA)

1 − cos(θB − θA) (rA − rD) ( ˆ T

r,θ

| θB − ˆ T | r,θ

θA )

− (θB − θA) (rA − rD)

ˆ

T | θ,θ

θB + ˆ T | θ,θ θA

(4.10)

|Ωi| ∂t V i,θ + (θB − θA) (rA T ˆ | r,θ

rA − rD T ˆ | r,θ

rD ) + θ B − θA

2

sin(θB − θA)

1 − cos(θB − θA) (rA − rD) ( ˆ T

θ,θ

| θB − ˆ T | θ,θ

θA )

+ (θB − θA) (rA − rD)

ˆ

T | r,θ

θB + ˆ T | r,θ θA

(4.11)

Equations(4.10)-(4.11) be asresultsof integrationo erΩi followedb pro ontoei,r

andei,θ of (4.7) Thisoperationisreferredtoasintegration-pro pro

Itis venientto equations(4.10)-(4.11)withtheresultofthetraditional h,thatispro

integrationpro appliedto(4.7) Thepro of (4.7)ontoer andeθ leadsto,

and

Equations(4.12)-(4.13)arenolonger ative theyownrighthandside terms,the ative

of the original equation (4.7) is lost during the pro operation This is due to variations of

v e r and e θ with resp to θ This kindof termsdoesnotappear ifCartesian ordinates are

in lieuofpolar ordinates

Now,integrating(4.12)-(4.13)yield,

|Ωi| ∂tVi,r + (θB − θA) (rA T ˆ | r,r

rA − rD T ˆ | r,r

rD ) + (rA − rD) ( ˆ T | r,θ

θB − ˆ T | r,θ

θA ) = Z

Ω i

T θ,θ (r, θ) dr dθ , (4.14)

|Ωi| ∂t V i,θ + (θB − θA) (rA T ˆ | r,θ

rA − rD T ˆ | r,θ

rD ) + (rA − rD) ( ˆ T | θ,θ

θB − ˆ T | θ,θ

θA ) = −

Z

Ω i

T r,θ (r, θ) dr dθ (4.15) The ofequation(4.10)with(4.14), and(4.11)with(4.15)issummarizedin thefollowingresult

Proposition 1 The integration-proje proedure andproje gration operation applied tove

equationwrittenin2D polar oordinatesareequivalentifandonlyif thesour etermsare etizedasfollows

Z

Ω i

T θ,θ (r, θ) dr dθ = (θB − θA) (rA − rD)

ˆ

T | θ,θ

θB + ˆ T | θ,θ θA 2 + 1 − θ B − θA

2

sin(θB − θA)

1 − cos(θB − θA)

! (rA − rD) ( ˆ T | r,θ

θB − ˆ T | r,θ

θA ) ,

Z

Ω i

T r,θ (r, θ) dr dθ = (θB − θA) (rA − rD)

ˆ

T | r,θ

θB + ˆ T | r,θ θA 2

− 1 − θB − θA

2

sin(θB − θA)

1 − cos(θB − θA)

! (rA − rD) ( ˆ T | θ,θ

θB − ˆ T | θ,θ

θA )

(4.16)

Note that the versionof h term be split into a tered (traditional)term and a

eterm Theexpressionofthis termisnewto thebestofourknowledge Notesp that

it the onentsofthetensorT

4.2 Tools for implementation

Inthis wegathertogether tools that areimportant for implementationof apartor ofthe

fullmodelsystem osedofa tinuityandmomentumequations



∂ t n + ∇ (nV ) = 0 ,

Here, nis the densityperunit mass, V is the velo y and I the unit tensor Both theCartesianversionof system(4.17) and its terpart in 2D polar ordinatesare investigated, andthe resultsobtainedb both methodsare

The ofthemeshusedinCartesian ordinatesarequadrilateralswhiletheyare edonesin2Dpolar ordinatessystem Inthetwo weusethesamenodesto themesh,butweemphasizethatthe

aredierentfromone ordinatessystemto another InFigure2aredisplayedtwo hkindsofmeshes

where the numberof radial Nr = 3 whilethose of azimuthal ones is Nθ = 6 The radialand azimuthal mesh stepsaredenoted b ∆rand∆θ ,sothatforauniformmeshin azimuthal onehas

∆θ = 2 π

N θ

Fig2: Cartesianand polarmeshes

Innitevolumemethodimplementation,weneedtoknowthevalueofareasofmesh Considera

Ωi in2D polar ordinateslo b itsfour nodesA, B, C, andD asinFigure1,themeasure ofthis areaisgivenb ,

|Ωi|r,θ =



r + ∆r 2



Ifthesenodesareusedto a ΩiinaCartesian ordinates,themeasureoftheareaofthis will be

|Ωi|x,y =



r + ∆r 2



Now,b taking∆θ smalli.e ∆θ → 0inequation(4.19),onegets

|Ωi|r,θ ≈ |Ωi|x,y ,

hmakesobviousthe thatforlargeN θ,meshesobtainedinCartesianand2Dpolar ordinatessystems

Next,weareinterestedinevaluatingtheintegralofnormalv alongedgesofmesh The kyone

seemstobethose ondingto ededgesasshownin Figure3 Thankstothediv theorem,

I

n dl = 0 ,

we

Z d AB

n dl = Z

AB

n dl ,

h beimmediately b knowningonlythe ordinatesofthenodesAandB

A

B

Fig3: Normals

Wearenow withthe ofn uxes forhyperb equationswritteningeneral

ordinatesin2D.Assumewehaveinourhandan uxpro [3,5,9℄formore

details Thefollowingisapossiblealgorithmthatallowsusto an uxingeneral

ordinatesof Ωi andΩj:

• Write the v quantities of Ωi and Ωj to the orthogonal basis (nij , τ ij ) of the

in boundary ∂Ωij between the Ωi and Ωj, (nij being the outward pointing unit v normaltothe∂Ωij fromthe Ωitothe Ωj,τ ij isanunitv orthogonalton ij) Let

Ωi andΩj betheresultsofthisstep;

• Computetheux Φij with thestatesΩi and Ωj withresp to thein boundary ∂Ωij b using

a hosenn ux[3,5,9℄;

• Pro theuxΦij ontothe Ωi andΩj togetuxesΦi and Φj resp ely, to

Φi = (Φij · ei,r ) ei,r + (Φij · ei,θ) ei,θ ,

Φj = (Φij · ej,r) ej,r + (Φij · ej,θ) ej,θ ,

where(ei,r , e i,θ )isthea eragev basisin the Ωi,(ej,r , e j,θ)isthoseofthe Ωj

2D