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HAL Id: inria-00464799

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Submitted on 18 Mar 2010

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schemes for steady inviscid flow problems on hybrid

unstructured meshes

Remi Abgrall, Adam Larat, Mario Ricchiuto

To cite this version:

Remi Abgrall, Adam Larat, Mario Ricchiuto. Construction of very high order residual distribution

schemes for steady inviscid flow problems on hybrid unstructured meshes. [Research Report] RR-7236,

INRIA. 2010, pp.60. �inria-00464799�

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a p p o r t

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Thème NUM

Construction of very high order residual distribution

schemes for steady inviscid flow problems on hybrid

unstructured meshes

Rémi Abgrall — Adam Larat — Mario Ricchiuto

N° 7236

(3)
(4)

Centre de recherche INRIA Bordeaux – Sud Ouest

problems on hybrid unstru tured meshes

RémiAbgrall , Adam Larat ,Mario Ri hiuto

ThèmeNUMSystèmesnumériques

Équipes-ProjetsBa hus

Rapportdere her he n°7236Avril200960pages

Abstra t: In this paperwe onsider the veryhigh order approximation of

solutionsoftheEulerequations. Wepresentasystemati generalizationof the

ResidualDistributionmethodof[5℄toveryhighorderofa ura y,byextending

thepreliminaryworkdis ussedin[18℄tosystemsandhybridmeshes. Wepresent

extensivenumeri alvalidationforthethirdandfourthorder aseswithLagrange

nite elements. In parti ular, we demonstrate that we an both have a non

os illatory behavior, even for very strong sho ks and omplex ow patterns,

andtheexpe teda ura yonsmoothproblems.

Key-words: Veryhighorders hemesfor ompressibleuidme hani s,hybrid

(5)

ompressible sur des maillages non stru turés

hybrides

Résumé: Dans e rapport, nous onsidérons leproblèmedel'approximation

deséquationsd'Eulerauxmoyensdes hémasd'ordretrèsélevés. Nous

présen-tonsune généralisationsystématique des s hémas dé rits dans [5℄ permettant

de onstruiredess hémasd'ordre(très)élevéutilisantdesmaillagesnon

stru -turéshybrides. Onmontre queles hémaobtenuest stable, même dansle as

ded'é oulements ompliqués,etatteintee tivementlapré isionre her héesur

dessolutionsrégulières.

Mots- lés : S hémas ompa ts d'ordreélevé pour lamé anique des uides

(6)

Contents

1 Introdu tion 4

2 Mathemati alproblem 5

3 Veryhigh orderresidual distribution: generalprin iplesand thes alar ase 6

3.1 Introdu tion: dis reteunknownsanddis reteequations . . . 6

3.2 A ura y onstraints . . . 9

3.3 Monotoni itypreservation . . . 11

3.4 Gettinghigh ordera ura yand monotoni itypreservation . . . 13

3.5 Spuriousmodesanditerative onvergen e: anumeri alexampleanda ounterexample 15 3.5.1 Caseoftriangles. . . 15

3.5.2 Caseofquadrangles. . . 16

3.6 Convergentnonlinears hemes . . . 17

3.7 Summaryofthenals hemefors alarproblems. . . 20

4 Numeri al illustrationsfor the s alar ase. 20 5 Extension to systems 23 5.1 Therstorderbuildingblo k . . . 23

5.2 Controllingtheos illations . . . 24

5.3 Spuriousmodelteringpro edure. . . 25

5.4 Boundary onditions . . . 27

5.5 Summaryofthenals hemeforthesystem ase. . . 29

6 Numeri al resultsforsystems 30 6.1 A onve tionproblem . . . 30

6.2 Computation ofjet . . . 30

6.3 Subsoni examples . . . 30

6.3.1 Asubsoni example: owoverasphere . . . 30

6.3.2 Subsoni owovertwoalignedspheres . . . 31

6.4 A transoni NACA0012airfoils ase . . . 32

6.5 TheRinglebtest ase . . . 32

6.6 A more omplex ase. . . 33

7 Con lusion 34

A Metri properties of the leftand righteigenve tors 37

(7)

1 Introdu tion

Inthere entyears,therehasbeenastrongeorttodeveloprobustandhigher

order(

> 2

)s hemesfor hyperboli equations,su h astheEuler equations,on

unstru turedgrids.

ExamplesaretheENO/WENOs hemes[1,2℄andtheDis ontinuousGalerkin

s hemes[3℄. IntheENO/WENO ase,theequationsareapproximatedbya

-nitevolumes heme wheretheentries oftheuxareevaluatedbyahighorder

re onstru tion polynomial. The latter is obtainedfrom the ell-data that are

interpreted as approximation of the average value of the solution on ontrol

volumes. Inouropinion,themaindrawba kofthisapproa hisitsalgorithmi

omplexityandthenon ompa tnatureof the omputationalsten il: the

av-eragevalueof thesolutionin a ellisupdatedbyusing itsneighbors,and the

neighbors of neighbors, and so on, depending on theexpe ted a ura y. The

non ompa tnessofthesten ilisalsoaseriousdrawba kfortheparallelization

ofthe ode.

In the ase of DG s hemes, the solution is approximated by alo al

poly-nomial that is dis ontinuous a rossthe interfa eof the elementsof the mesh.

The solution is updated by means of a lo al Galerkin form of the equations.

The dis ontinuous nature of the representation requires the use of numeri al

uxeswhenintegrationbypartsisperformedontheuxdivergen eterm. The

DGapproa hinvolvesaverylo alformulation, andit is indeed quiteexible.

However,ithasonemaindrawba kinthefastgrowthofthenumberofdegrees

offreedom(seealsothedis ussionin [4℄).

Inthispaperwehave hosentouseadierentstrategybasedontheResidual

Distribution(RD)approa hof[5℄. IntheRDmethod,thesten ilisverylo al,

asinDG,butthenumberofdegreesoffreedomgrowslessqui kly. Thepri eto

payistoimposethe ontinuityoftheapproximation(seehowever[6,7℄),asin

standardniteelementmethods. Indeed,theRDs hemes anbeseenasnite

elements where the test fun tions may depend on the solution. This lass of

s hemeishavingagrowinginterest(see [8,9,10,11, 12,13,14, 15,5, 16,17℄,

et .). Mostoftheexistingwork,however,islimitedtose ondorderofa ura y,

with the ex eption of the work dis ussed in [15, 14, 17℄, and, more re ently,

in [18℄. In this paper, weextendthe preliminary resultspresentedin the last

referen ebydis ussingtheirappli ationtothe aseoftheEulerequation,andby

presentinganextensivenumeri alevaluationoftheperforman eofthes hemes.

Thestru tureofthepaperisasfollows. Inse tionŸ3were allthe

onstru -tionofveryhigh orderRD s hemes,followingthepreliminary work presented

in [18℄ that we extendhereto hybridmeshes. Starting from the generalform

oftheRDdis retization,weintrodu ethe onditionsleadingtoveryhighorder

of a ura y and monotoni ity, and nally presentthe basi onstru tion used

in the paper. Some s alar numeri al tests are also dis ussed to demonstrate

thevalidityoftheapproa hdes ribed. Theextensionofthes hemesto

hyper-boli systems,andinparti ularto theEuler equations,istheobje tofse tion

Ÿ4. Allthedis retization stepsdes ribedinse tionŸ3arerevisited anddetails

(8)

presented in se tion Ÿ5. The paper is ended with some on lusiveremarks, a

summaryofthefutureandongoingdevelopmentsoftheworkpresentedhere.

2 Mathemati al problem

Weareinterestedinthenumeri alapproximationofsteadyhyperboli problems

oftheform

div

f

(u) = S(u)

(1a)

whi haredenedonanopenset

Ω ⊂ R

d

,

d = 2, 3

withweakDiri hletboundary

onditions,

u = g.

(1b)

denedontheinowboundary

1

∂Ω

= {x ∈ ∂Ω, ~n · ∇

u

f

< 0}.

In(1),theve torofunknown

u

belongsto

R

p

,andtheux

f

is

f

= (f

1

, . . . , f

d

).

In(1a),

S

isasour etermwhi hhereonlydepends ontheunknown

u

.

The main target example we are interested in is the system of the Euler

equationswith theve torofunknown

u

= (ρ, ρ ~u, E)

T

where

ρ

isthedensity,

~u

is thelo al owspeed, and

E

isthetotal energy. In

theparti ular ase

d = 2

,setting

~u = (u

1

, u

2

)

T

,theux anbewrittenas

f

1

=

ρu

1

ρu

2

1

+ p

ρu

1

u

2

u

1

(E + p)

 ,

f

2

=

ρu

2

ρu

1

u

2

ρu

2

2

+ p

u

2

(E + p)

 .

(2)

Thesystemis losed by anequation of statethat relates thepressure

p

to

u

.

Hereweassumeaperfe tgasequationofstate,

p = (γ − 1)



E −

1

2

ρ||~u||

2



with

γ = 1.4

. 1

~

n

(9)

3 Very high order residual distribution: general

prin iples and the s alar ase

3.1 Introdu tion : dis rete unknowns and dis rete

equa-tions

Let

τ

h

denoteatessellationofthespatialdomain

. Inthispaper

τ

h

isassumed

tobe omposedoftrianglesandquadsin2D

2

.Ageneri elementisdenotedby

K

. Denote by

n

t

thenumberofelementsofthemesh. Themesh parameter

h

denotesthemaximum radiusoftheouter ir lesof theelements

K ∈ τ

h

. The

verti esofthemesharedenotedby

{M

i

}

i=1,...,n

s

. Whenthereisnoambiguity,

wedenotetheverti esofanelement

K

by

1, . . . , n

K

d

.

Inourapproa h,thedis reteunknownsareasetoflo alvaluesofthesolution

insomemeshlo ations,su h ase.g. theverti es

M

i

, edgemid-pointset .,et .

Theseunknowns arereferredto astheDegreesofFreedom(DOF). Denoteby

l

}

l=1....,n

dof

thelist ofdegreesoffreedom. Inthe aseofase ondorder RD

s heme, the DOF are the verti es of the mesh, that is :

σ

l

= M

l

, ∀ l

. To

onstru tahigherordera urateRD s heme,therearetwooptions:

1. The ontributiontothedis reteequationofaDOF

σ

l

inageneri element

K

is obtained by using information outside

K

. This option has been

followed in [19℄, and in [20, 14℄. In this ase, the ompa tness of the

omputationalsten ilisredu ed,withthemaindrawba kofanin reased

algorithmi omplexity,espe iallywhenmorethanthirdorderofa ura y

issoughtfor.

2. Dis reteequationsarewrittenin anelementbyelementfashion, without

using anyinput outsideea h element. Naturally,in this ase, additional

DOFs need to be storedin ea h element, in orderto beable to in rease

thea ura y. Thisistheapproa hfollowede.g. in[21,18℄.

Here, following our initial work [18℄, we use a lo al higher order polynomial

interpolation allowing to keep the lo al element-by-element stru ture of the

RDformulation. Severalwaysofobtaining ontinuous

k

-thdegreepolynomials

exist. Inthis paper,wewillfo usonthe aseofstandard

P

k

and

Q

k

Lagrange

elementsdenedasfollows:

ˆ Quadrati interpolation: theDOFsarethesolutionvaluesintheverti es

andtheedgesmidpoints. Thisyields

3 + 3

pointspertrianglein2Dand

4 + 6

pointspertetrahedronin3D. Foraquadrangle,weneedtoaddthe

entroid,leadingto

4 + 5

points perelementsin 2D.The3D ase would

need

27

DOFsperelement.

2

Wehaveobtainedresultsin3D,notreportedinthispaper. Wehavenotyet onsidered

the aseofhybridmeshesin3D,eventhoughourmethodshouldextendwithoutproblemsto

(10)

ˆ Cubi interpolation: in the 2D ase,the DOF are the verti es, 2points

peredge (whi h with the verti es form three segments of equal length),

and the entroid, i.e.

3 + 2 × 3 + 1

DOF per element. In3D ase, the

DOF arethe 4verti es,2 DOFperedge, and the entroidof ea h fa e,

i.e.

4 + 6 × 2 + 4 = 20

DOFs. The aseofquadrangleelementsleadsto

16

DOFperelements,the3D asewouldneed64DOFsperelements.

ˆ et .

Notethatthe ontinuityofthestandardLagrangeelementsrequiresthatallthe

DOFonelementboundariesaresharedbyneighboringelements.

Asa onsequen e,inthetriangular/tet ase,we an ountthetotalnumber

ofDOFin termsof thenumberofverti es,edges,fa es(in3D)andelements,

ˆ inthe2D ase,wehave

 Quadrati :

n

s

+ n

edge

DOFs,

 Cubi :

n

s

+ 2n

edge

+ n

t

DOFs.

ˆ inthe3D ase,ifin addition

n

f ace

isthenumberoffa es, wehave

 Quadrati :

n

s

+ n

edge

DOFs,

 Cubi :

n

s

+ 2n

edge

+ n

f ace

DOFs.

Thanks to the Euler formula, it is possible to give, for aregular

trian-gulation,anestimateof theasymptoti behavioroftheglobalnumberof

DOF. It is known that in 2D, we have

n

edge

≈ 3n

s

and

n

t

≈ 2n

s

and

in 3D,

n

edge

≈ 7n

s

,

n

f ace

≈ 10n

s

and

n

t

≈ 6n

s

. On Table1, we have

reportedtheasymptoti numberofdegreesoffreedomwithrespe ttothe

dimension and the degree of interpolation. Forthe sake of omparison,

we have also given the same parameters in the ase of a dis ontinuous

approximation, asthe oneused in DG s hemes. It is lear that the

on-tinuous approximationrequires amu h smallernumberof DOFto yield

the same polynomial representation, but the number of DOF in rease

morerapidlyfor ontinousapproximationsthan dis ontinousones. Both

asesare asymptoti allysimilar. Thesame on lusion alsoholds for the

quad/hex.

On ewehaveestablishedwhatourdis reteunknownsare,wehavetoprovide

ea hofthemwithadis reteequation. Wedistinguishtwo ases.

1. in the aseof an internal DOF

σ

, aresidual distribution s hemefor (1)

reads

forall

σ ∈ τ

h

,

X

T ∋σ

Φ

K

σ

= 0 ,

(3)

wherethesplitresiduals

Φ

K

σ

in(3)mustsatisfythefollowing onservation

onstraint forany

K,

X

σ∈K

Φ

K

σ

=

I

∂K

f

h

(u

h

) · ~ndl −

Z

K

S

h

(u

h

)dx := Φ

K

(4)

(11)

where

f

h

(u

h

)

and

S

h

(u

h

)

arehigh order a urateapproximationsof the

ux

f

(u)

and the sour e term

S(u)

. Natural hoi es are: the Lagrange

interpolantof

f

(u)

atthedegreesoffreedomdening

u

h

,orthetrueux

evaluatedfor

u

h

.

2. if

σ

is a DOF lying on the boundary of

, the equation for

σ

has to

take into a ount the boundary onditions. Let

Γ

be any edge/fa e of

theinowboundaryof

. We onsideranumeri alux

F

whi hdepends

on theboundary ondition

u

, theinwardnormal

~n

and the lo al state

u

h

. Then we dene boundary residuals

Φ

Γ

σ

whi h satisfy the following

onservationrelation forany

Γ ⊂ ∂Ω,

X

σ∈Γ

Φ

Γ

σ

=

Z

∂Γ



F (u

h

, u

, ~n) − f

h

(u

h

) · ~n



dl := Φ

Γ

,

(5)

Atthispointwe anwriteforanarbitraryDOFon

∂Ω

:

forall

σ ∈ ∂Ω,

X

K∋σ

Φ

K

σ

+

X

Γ⊂∂Ω

,Γ∋σ

Φ

Γ

σ

= 0.

(6)

Thenfollowing[21℄,itiseasytoshowthatifthesequen e

u

h

isboundedin

L

when

h → 0

, and if there exists

v

su h that

u

h

→ v

when

h → 0

, then

v

is

aweak solutionof (1). Oneessentialingredientof theproofis the ontinuity

oftheinterpolanta rossedges. One anhoweveralleviatethis onstraint,and

deneRDs hemesondis ontinuouselements,see[22,6,7℄forthese ondorder

ase. Additional onstraints, su h asthe satisfa tionof anentropyinequality,

ouldbesetbut thiswillnotbe onsideredinthis paper.

Remark3.1 (Numeri alquadrature). Before going further, letus make a

re-mark on erning the notation, andthe denition of the element andboundary

edge residuals

Φ

K

, and

Φ

Γ

,respe tively. The denitions (4)and, (5) a priori,

needexa t integration of thedis reteuxandsour e. However, inpra ti e

nu-meri alquadratureismoreoftenimplemented. Inthis ase,werepla e (4)and

(5)by for any

K,

X

σ∈K

Φ

K

σ

=

X

e∈∂K

|e|

G f

X

p=1

ω

p

f

h

(u

h

(x

p

))·~n

e

−|K|

Gv

X

p=1

ω

p

S

h

(u

h

(x

p

)) := g

Φ

K

.

(7a)

havingdenotedby

e

the generi edge (fa e in3d) of

K

,and

for any

Γ ⊂ ∂Ω

,

|Γ|

G f

X

p=1

ω

p



F (u

h

(x

p

), u

(x

p

), ~n

Γ

)−f

h

(u

h

(x

p

))·~n

Γ



:= f

Φ

Γ

(7b) wherethe G f and G v

denote the number of fa e andvolume Gauss pointsused

in the numeri al quadrature. The hoi e of G

f

and G

v

, viz of the quadrature

formulasusedinpra ti e, shouldnotdegradethe a ura yof thedis retization.

(12)

3.2 A ura y onstraints

In the previous se tion we have introdu ed the general abstra t form of our

RDdis retization. Thisformulationinvolvesintegralsofnumeri al

approxima-tionof the uxes (andof the sour eterm) basedon the

P

k

and

Q

k

Lagrange

approximation of the unknown

u

h

. These integralsare in pra ti eevaluated

numeri ally, and repla ed eventually by the quadrature integralsof equations

(7a)and(7b) .

The a ura y obtained in pra ti e is of ourse dependent on the type of

quadratureusedintheimplementationofthes hemes. Inorderto hara terize

thisdependen e,wefollowthe trun ationerroranalysis of[21℄. Followingthe

lastreferen e, one anshowthat s heme(3), (6) , (4)and (5)satises, forany

ϕ ∈ C

1

0

(Ω)

3

,thefollowingtrun ationerror

E(w

h

, ϕ

h

) =

X

σ∈Ω

ϕ(σ)

X

K∋σ

Φ

K

σ

+

X

Γ⊂∂Ω

,Γ∋σ

Φ

Γ

σ

!

=

Z



div

f

h

(w

h

) − S

h

(w

h

)



ϕ

h

(x) dx +

X

K⊂Ω

1

#{σ ∈ K}

X

σ,σ

K

ϕ(σ) − ϕ(σ

)



Φ

K

σ

− Φ

K,c

σ



+

Z

∂Ω



F (w

h

, w

, ~n) − f

h

(w

h

) · ~n



ϕ

h

(x) dl +

X

Γ⊂∂Ω

1

#{σ ∈ Γ}

X

σ,σ

Γ

ϕ(σ) − ϕ(σ

)



Φ

Γ

σ

− Φ

Γ,c

σ



= −

Z

∇ϕ

h

(x) · f

h

(w

h

) +

Z

∂Ω

ϕ

h

(x)f

h

(w

h

) · ~ndl −

Z

ϕ

h

(x)S

h

(w

h

)dx

+

Z

∂Ω



F (w

h

, w

, ~n) − f

h

(w

h

) · ~n



ϕ

h

(x)dl

+

X

K⊂Ω

1

#{σ ∈ T }

X

σ,σ

K

ϕ(σ) − ϕ(σ

)



Φ

K

σ

− Φ

K,c

σ



+

X

Γ⊂∂Ω

1

#{σ ∈ Γ}

X

σ,σ

Γ

ϕ(σ) − ϕ(σ

)



Φ

Γ

σ

− Φ

Γ,c

σ



.

(8)

where

w

is a lassi al solution of the problem,

w

h

being its

P

k

/

Q

k

Lagrange

approximation,

ϕ

h

is theLagrange interpolant of

{ϕ(σ)}

σ

,

Φ

K,c

σ

and

Φ

Γ,c

σ

are

theGalerkinresiduals

Φ

K,c

σ

=

Z

K

ψ

σ



div

f

(w

h

)−S(w

h

)



dx

and

Φ

Γ,c

σ

=

Z

Γ

ψ

σ



F (w

h

, w

, ~n)−f (w

h

)·~n



dx ,

(9) and

ψ

σ

∈ P

k

(K)

or

Q

k

(K)

is theLagrangebasisfun tion relativetotheDOF

σ ∈ K

. Using thefa tthat everywherediv

f

(w) − S(w) = 0

,theerror anbe

3

C

1

0

(Ω)

isthesetof

C

1

(13)

easilyde omposedas[21,23℄

E(w

h

, ϕ

h

) = −

Z

∇ϕ

h

(x) · f

h

(w

h

) − f (w)



(

uxapproximationerror

)

Z

ϕ

h

(x) S

h

(w

h

) − S(w)



dx

(

sour eapproximationerror

)

+

Z

∂Ω



F (w

h

, w

, ~n) − f (w) · ~n



ϕ

h

(x)dl

(

BC approximationerror

)

+

X

K⊂Ω

1

#{σ ∈ K}

X

σ,σ

∈K

ϕ(σ) − ϕ(σ

)



Φ

K

σ

− Φ

K,c

σ



(

distribution error-interior

)

+

X

Γ⊂∂Ω

1

#{σ ∈ Γ}

X

σ,σ

Γ

ϕ(σ) − ϕ(σ

)



Φ

Γ

σ

− Φ

Γ,c

σ



(

distribution error-boundary

).

(10)

Asdis ussedthoroughlyin[21,23℄,relation(8)isa onsequen eofthe

onser-vationrelations (4)and (5). Followingagain thelast referen es,we anprove

thefollowingresult:

Proposition3.2. Given aregularenough lassi al solution

w

,ifthe residuals

evaluatedon the

P

k

, Q

k

interpolant

w

h

satisfy

Φ

K

σ

(w

h

) = O(h

k+d

)

(11a) and

Φ

Γ

σ

(w

h

) = O(h

k+d−1

),

(11b)

andif the approximations

f

h

(w

h

)

, and

S

h

(w

h

)

are

k + 1

-order a urate, then

the trun ationerrorsatises

|E(w

h

, ϕ

h

)| ≤ C(ϕ, f , w) h

k+1

.

The onstant

C(ϕ, f , w)

depends onlyon

ϕ

,

f

,and

w

.

Proof. Seeappendix B.

A rst onsequen e of the analysis is obtained by noting that, under the

hypothesesofproposition3.2(see[21, 23℄fordetails) :

Φ

K

(w

h

) = O(h

k+d

)

and

Φ

Γ

(w

h

) = O(h

k+d−1

)

Asa onsequen e,ifthere existsa onstant(inthes alar ase)oramatrix(in

thesystem ase)

β

T

σ

su h that

Φ

K

σ

= β

K

σ

Φ

K

(12a)

(14)

thenthe ondition(11)isfullledprovidedthat

β

T

σ

isuniformlybounded. This

gives a design riterion for high order s hemes. For histori al reasons, RD

dis retizations that anbe written asin (12a)-(12b) are referred to Linearity

Preserving even-thoughtheinterpolantisnolongerlinear.

Inpra ti e,asexplainedinremark3.1,oneusesofnumeri alquadratureto

evaluate the ell residuals. In this ase, linearitypreserving RD s hemes read

( f. equations(7a)and(7b) )

Φ

K

σ

= β

K

σ

g

Φ

K

(13a)

Φ

Γ

σ

= β

σ

Γ

Φ

f

Γ

(13b)

To maintainthe sameerror level, wesee immediately that the onstraintson

thequadratureformulasusedtoobtain(7a)and(7b) arethefollowing:

ˆ In(13a),wemusthave

X

e∈∂T

|e|

Gf

X

p=1

ω

p

f

h

(u

h

(x

p

)) · ~n

e

=

I

∂T

f

h

(u

h

) · ~ndl + O(h

k+d

)

(14a) and

|T |

Gv

X

p=1

ω

p

S

h

(u

h

(x

p

)) =

Z

T

S

h

(u

h

)dx + O(h

k+d

)

(14b)

ˆ In(13b) ,wemusthavefortheboundary onditionsintegrals

|Γ|

Gf

X

p=1

ω

p



F (u

h

(x

p

), u

(x

p

), ~n

Γ

) − f

h

(u

h

(x

p

)) · ~n

Γ



=

Z

∂Γ



F (u

h

, u

, ~n) − f

h

(u

h

) · ~n



dl + O(h

k+d−1

)

(14 )

There are of ourse numerous quadrature formulas that an be used to stay

within the error bounds given above. An inferior bound to the polynomial

degreethat hastobeintegratedexa tlyisgivenbythesebounds.

Thepra ti alapproa husedinthisworkistore onstru tinea helementa

uxpolynomialbasedontheLagrangeinterpolationoftheuxvaluesevaluated

atthedegreesoffreedom. Thequadraturepoints oin ide withtheDOF,and

thequadratureweightsareeasily omputedon eandforall. Thisisequivalent

toaquadraturefreeapproa h( f. [24℄). We ome ba ktothispointinse tion

5.4todis ussoura tualimplementationoftheboundary onditions.

3.3 Monotoni ity preservation

Inthisparagraphwe onsidertheissueofguaranteeingthenon-os illatory

(15)

[25℄todesigndis retizationsyieldingsolutionsthatverifyadis retemaximum

prin iple. To do this, we onsider the ase in whi h (1) is a s alar equation

forthe unknown

u

, and the homogeneous ase

S = 0

. Inthis setting, all RD

s hemes anbere-writtenas

Φ

K

σ

=

X

σ

∈T

c

σσ

(u

σ

− u

σ

)

(15)

sothatequation(3)be omesforany

σ

(andnegle tingboundary onditions)

X

T ∋σ

X

σ

T

c

σσ

(u

σ

− u

σ

) = 0.

Ingeneral,the oe ients

c

σσ

depend on thesolution, whi h meansthat the

lastexpressiondenes aset ofnon linearequations thatneedsto besolvedby

meansofaniterativemethod. ThesimplestoneistheJa obilike iteration

u

n+1

σ

= u

n

σ

− ω

σ

X

K∋σ

X

σ

∈T

c

K

σσ

(u

σ

− u

σ

)

!

n

(16)

where

ω

σ

isarelaxationparameter.

Itiseasytoverifythatifthes hemesatisesthepositivity onditions

X

K∋σ,K

∋σ

c

K

σσ

≥ 0 ∀ σ, σ

and

1 − ω

σ

X

K∋σ

X

σ

∈K

c

K

σσ

!

≥ 0

∀ σ

(17)

thenthesolutionveriesthefollowingdis retemaximumprin iple

min

K∋σ

σ

min

K

u

0

σ

≤ u

n

σ

≤ max

K∋σ

σ

max

K

u

0

σ

(18) havingdenoted by

u

0

σ

thevaluesoftheinitialsolutionintheDOFlo ations. A

more onvenientapproa hforthedesignofthes hemesistorepla e onditions

(17) bylo al onstraints. Todothis, letusdene forea h DOFthefollowing

mediandualareas:

C

σ

K

=

|K|

n

d

and

C

σ

=

X

K∋σ

C

σ

K

Obviously, onditions(17)aremetifthefollowinglo alpositivity onditionsare

veried:

c

K

σσ

≥ 0 ∀ σ, σ

∈ K

and

∀ K

and

ω

σ

max

K∋σ

"

C

σ

C

K

σ

 X

σ

∈K

c

K

σσ

#

≤ 1 ∀ σ

(19)

These onditionsdonotimplythattheiteratives heme(16)is onvergent,but

onlythat the dis retemaximum prin iple(18)(viz.

L

-stability)is satised.

In the rest of the paper, a s heme that veries onditions (19) is said to be

(16)

3.4 Gettinghighordera ura yand monotoni ity

preser-vation

Itisknownthatas hemethatismonotoni itypreservingwith oe ients

c

K

σσ

independent on the solution annot satisfy (11). This results is a variant of

Godunov's theoremfor RD s hemes,of whi h ageneralproof isbeengiven in

[26℄. As a onsequen eofthis, amonotoni ityand linearitypreservings heme

mustbenon linear. Here wefollow[21℄ to obtainnonlinears hemes verifying

bothproperties.

Westartfromamonotonerstorders hemewhi hresidualsare(for

S = 0

)

Φ

L

σ

=

X

σ

K

c

L

σσ

(u

σ

− u

σ

)

thesuper-s ript

L

standingforLoworder. Byassumption,the oe ients

c

L

σσ

areallpositive,and,of ourse,

X

σ∈K

Φ

L

σ

= Φ

K

Then,let

Φ

H

σ

denotehighorderresiduals,su hthat

Φ

H

σ

= β

σ

Φ

K

with

X

σ∈K

β

σ

H

= 1

(20)

Byanalogy,weintrodu etheparameters

x

σ

dened by

x

σ

=

Φ

L

σ

Φ

K

forwhi h,thankstothe onservationrelation,wealsohave

P

σ

x

σ

= 1

.

Thenextstepistowrite theformalidentity

Φ

H

σ

=

Φ

H

σ

Φ

L

σ

Φ

L

σ

=

X

σ

Φ

H

σ

Φ

L

σ

c

L

σσ

(u

σ

− u

σ

)

andwe anseethat,startingfrom

Φ

L

σ

,we ouldobtainamonotoni ity

preserv-inghighorders heme,providedthatwesatisfythe onstraint

Φ

H

σ

Φ

L

σ

≥ 0

∀ σ ∈ K

be ausethenwehaveforthehighorders heme

c

H

σσ

=

Φ

H

σ

Φ

L

σ

c

L

σσ

≥ 0

(17)

1. Conservation.

X

σ∈K

β

σ

= 1

and

X

σ∈K

x

σ

= 1

2. Monotoni itypreservation.

x

σ

β

σ

≥ 0

∀ σ ∈ K

These relations an beinterpretedgeometri ally. Sin e there is noambiguity,

we an assumethat thedegreesoffreedom anbenumberedfrom

1

to

n

d

,an

weidentifytheDOF

σ

toitsnumber

in

[1, . . . , n

d

]

.

Let us onsider in

R

n

d

n

d

linearly independent points

S = {A

}

ℓ=1,··· ,n

d

.

Notethey donot have onne tionswith any physi al points in themesh. We

anintrodu eforanypoint

M ∈ R

n

d

itsbary entri oordinates

(M )}

with

respe tto

S

:

M =

n

d

X

ℓ=1

λ

(M )A

orequivalently,forany

O ∈ R

n

d

−−→

OM =

n

d

X

ℓ=1

λ

(M )

−−→

OA

.

Wehavebydenition

n

d

P

ℓ=1

λ

(M ) = 1

. Thus,we aninterpret

{x

l

}

and

l

}

as

thebary entri oordinatesofthepoints

L

and

H

su hthat

L =

n

d

X

ℓ=1

x

A

H =

n

d

X

ℓ=1

β

A

andtheproblembe omestodeneamappingonto

R

n

d

: L 7→ H

su hthat the

onstraints

x

β

≥ 0

aretrue: theadvantageofthat interpretationisthat the

onservationpropertiesareautomati allysatised.

Therearemanysolutiontothat problem,oneparti ularlysimpleoneisan

extensionofthePSIlimiter ofStruijs:

β

=

x

+

X

x

+

,

x

+

= max(x, 0).

(21)

Thereisnosingularityintheformulasin e

X

x

+

=

X

x

X

x

≥ 1.

(18)

3.5 Spurious modes and iterative onvergen e : a

numer-i al example and a ounter example

Before pro eeding further with the onstru tion, we onsider two numeri al

examplesinvolvingthesolutionoftheadve tionequation

~λ · ∇u = 0,

x ∈ Ω.

(22) 3.5.1 Case of triangles. Usingthe

P

k

interpolantin

T

u

h

=

X

σ∈T

u

σ

ψ

σ

,

thetotalresidual

Φ

T

anbewrittenas

Φ

T

=

Z

T

~λ · ∇u

h

dx =

X

σ∈T

u

σ

Z

T

~λ · ∇ψ

σ

dx.

Byanalogywithwhat isdonewithse ond orderRDs hemes[11,27℄, weset

k

σ

=

Z

T

~λ · ∇ψ

σ

dx,

sothat

Φ

T

=

X

σ∈T

u

σ

k

σ

.

(23) Wenote that

P

σ∈T

k

σ

= 0

. To onstru tanonlinears heme, westart from the

followingrstorder (lo al)Lax-Friedri h's(LLxF)s heme:

Φ

T

σ

=

Φ

T

n

d

+ α

T

(u

σ

− ¯

u)

(24a) with

¯

u =

X

σ

T

u

σ

n

d

.

(24b)

Using(23) ,theLLxF residual(24a) anbere astin theform(15)with

c

T

σσ

=

k

σ

− α

T

n

d

and

c

T

σσ

≥ 0

if

α

T

≥ max

σ∈T

|k

σ

|.

(25)

This rst order s heme is extremely dissipative, but this is the one from

(19)

systems, and it easily generalizes to any order of a ura y (viz. polynomial

interpolation).

WetestthenonlinearlimitedLLxFs hemeobtainedbyapplying(21)tothe

LLxF s heme (24a) s heme on twosimple linearadve tion problems. On the

spatialdomain

Ω = [0, 1]

2

,wetakeintherstproblem

~λ = (1, 2)

T

and

u(x, y) =



1

if

x = 0

and

y > 0

0

if

y = 0

and

x > 0

(26)

These ond problemisobtainedbysetting

~λ = (y, −x)

T

and

u(x, y) =



ϕ

0

(x)

if

y = 0

0

otherwise (27) where

ϕ

0

(x) =



cos

2

(2πx)

if

x ∈ [0.25, 0.75]

0

else

The results obtained in the

P

2

ase are displayed on gure 1. The behavior

observedis similar to what hasbeen found, in the se ond order ase, in [5℄ :

dis ontinuities are approximated without over- or undershoots, however, we

observethe appearan e of plateaus in the numeri al results, for both smooth

andnon-smoothdata. This is learlyvisible in theplotsoftheoutlet dataon

gure1. Othersymptomsare: smoothsolutionsoftenpresentahighfrequen y

os illations(spurious modes), the iterative onvergen e is poor: after avery

qui kdropofabouttwoordersofmagnitude,theiterativeresidualstagnatesto

a onstantvalue. Thebehavioris thesameobservedin nite volumes hemes

when using an over- ompressivelimiter. As remarked in [5℄, this behavior is

notdue to an

L

instability : the lo al maximum prin iple is satised both

theoreti allyandnumeri ally.

As a resultof thela k of iterative onvergen e,equation (3) is notsolved

exa tlybut within a

O(h)

error whi h anbeeasily measuredexperimentally

[5℄. Thustheoveralla ura yobtainedisonlythatofarstorder s heme.

3.5.2 Case of quadrangles.

This example is maybe more illuminating sin e we an exhibit some of the

spuriousmodes. Again,we onsider

Ω = [0, 1]

2

whi hisdis retisedbyuniform

quads. Theverti esare

x

i,j

= (

i

N

,

j

N

)

(

0 ≤ i, j ≤ N

)andtheproblemwrites

∂u

∂x

= 0

subje tedtoboundary onditionsontheleftsideof

. AssumingageneralLP

s heme,weupdatethesolutionby

u

n+1

σ

= u

n

σ

− ω

 X

K,σ∈K

β

σ

K

Φ

K

(20)

Twothingsneedto bepre ised: theboundary onditionsandtheinitial state

u

0

. Ontheleftboundary(inow),weimposea he k-boardlikemode,butthis

isnotreallyessentialasweseeat theend oftheparagraph),i.e.

u

σ

= (−1)

i

σ

where

i

is the indexsu h that x

σ

= (

i

N

, 0)

and

u

σ

= 0

is

σ

is anymid point.

Theinitial onditionisdened by

ˆ eitherasongure2-a: wepropagate theboundary onditionalongthe

hara teristi softhePDE.

ˆ orasongure2-b.

We expe t to onverge to the rst initialization. Let us ompute the total

residualon

Q = [x

i

, x

i+1

] × [y

j

, y

j+1

]

. Weget

Φ

Q

=

Z

y

j+1

y

j

Z

x

i+1

x

i

∂u

∂x

dxdy =

Z

y

j+1

y

j

u(x

i+1

, y) − u(x

i

, y)



dx.

Inour ase, wehave, bysymmetry,

u(x

i+1

, y) = u(x

i

, y)

, so that

Φ

Q

= 0

and

u

n+1

σ

= u

n

σ

. Thisshowsthatthes heme annot onvergeinthis ase.... Hen e,

somethingmoremustbedone!

3.6 Convergent nonlinear s hemes

Followingour previouswork [5,18℄, the behaviordes ribed in thelast se tion

anbe orre tedbyaddingtotheresiduals(13)atermoftheform

h

K

Z

K



~λ · ∇ψ

σ



τ



~λ · ∇u

h



dx,

τ > 0.

(28)

Thisisastreamlinedissipationterm,usedinSUPGdis retizationsofhyperboli

problems to suppress the spurious modes of theGalerkin s heme [28℄. As its

namesuggests,thistermhasadissipativenature. Itdoesnotdestroytheformal

a ura y of the original dis retization. It does not destroy the onservation

property(4)be ausetheresidualsarenow

Φ

σ

= Φ

H

σ

+ h

K

Z

K



~λ · ∇ψ

σ



τ



~λ · ∇u

h



dx

(29) with

Φ

H

σ

denedby(20)and learly,

X

σ∈K

Φ

σ

= Φ

K

be ause

P

σ∈K

∇ψ

σ

= 0

.

In fa t, (28) has the good ee t of removing the spurious modes that are

(21)

Theproblem ofintegral(28)is thatits exa tevaluation requires theexa t

integration of a polynomial of degree

2(k − 1)

whi h is expensive. A better

analysis of the stru ture and the role of the dissipative term helps to redu e

substantiallyits omputational ost. Westartbyrewritingthenonlinearlimited

s hemeas

Φ

K

σ

= Φ

K,c

σ

+ Φ

K

σ

− Φ

K,c

σ

,

with

Φ

K,c

σ

still given by (9). Given a fun tion

ϕ

, multiply (3) by

ϕ(σ)

and

addupall theequationsfor alltheDOF ofthe mesh. Using the onservation

relations,andnegle tingboundary onditions,(3)isequivalentto

Z

ϕ

h

div

f

h

(u

h

)dx +

X

K

q

K

h

, u

h

) = 0

(30a) with

q

K

h

, u

h

) =

1

n

d

!

X

σ,σ

K

(ϕ(σ) − ϕ(σ

))



β

σ

K

Φ

K

− Φ

K,c

σ



(30b) with

Φ

K,c

σ

theGalerkinresiduals(9). Themodi ationintrodu edin[5℄amounts

toadding term(28)to thequadrati form

q

K

. Theproblem is toknowunder

whi h onditionstheresultings hemeisdissipative,keepstheoriginala ura y,

andpreservesthenonos illatorybehaviorof (30a) .

Themostnaturalwayofpro eedingistorepla e(28)byanapproximation

obtainedbymeansofaquadrature formula:

d

K

h

, u

h

) = |K|

X

x

quad

ω

quad

"

~λ · ∇ϕ

σ



(x

quad

) τ (x

quad

)



~λ · ∇u

h



(x

quad

)

#

(31) su hthat

h

, u

h

) 7→

Z

ϕ

h

div

f

h

(u

h

)dx +

X

K



q

K

h

, u

h

) + θ

K

h

K

d

K

h

, u

h

)



isdissipative. Wehaveput quadraturebetweenquotesbe auseasweseelater

inthetext, thesequadratureformuladonotneedtobe onsistentas

approx-imationof integrals.

Insteadofstudying theoverall behaviorofthes heme,we anmakeuseof

the ompa tnatureofthedis retizationandfo usourattentiononthequadrati

form

h

, u

h

) 7→ q

K

h

, u

h

) + θ

K

h

K

d

K

h

, u

h

)

Here,

h

K

is a the radius of the ir le/sphere ir ums ribed to

K

. The

pa-rameter

θ

K

has the role of a tivate the extra dissipation in smooth regions,

while dea tivating it a ross dis ontinuities, where the original s heme has no

under/overshoots. Hen e,weshouldhave

θ

K

≈ 1

in orresponden eofsmooth

variationsofthesolutions,while

θ

K

≪ 1

a rossdis ontinuities.

Con erning a ura y on smooth solutions, whatever quadrature formula

weuse, andthankstotheterm

θ

K

h

K

infrontof

d

K

,if

u

h

(22)

asmoothenoughfun tion su hthat

~λ · ∇u = 0

,thenone anshow(see[18℄for details)that

θ

K

h

K

d

K

h

, u

h

)

≤ C(u)||∇ϕ|| h

k+d+1

K

,

sothat theformal a ura y isnot spoiled. Lastly, to ensure onvergen e,one

anaskthebilinearform

h

, u

h

) 7→ d

K

h

, u

h

)

to be positivedenite whenever

~λ · ∇u

> 0

. Inparti ular, following[5, 18℄,

wewillrequirethat

ω

quad

> 0

andthat

d

K

h

, u

h

)

ispositivedenitewhenever

~λ · ∇u

h

6= 0

.

Thisamountsatrequiringanumberofquadraturepointsallowinganexa t

representation of the term

~λ · ∇u

h

over

K

. Hen e, one quadrature point is

enoughfor

k = 1

,3pointsareneededfor

k = 2

,6for

k = 3

andsoon: onehas

to be exa ton polynomialsof degree

k − 1

. Inthe 3D ase, onequadrature

pointisenoughfor

k = 1

,3for

k = 2

,andsoon.

There is no need for the quadrature formula to be onsistent with the

integral

Z

K

(~λ · ∇ϕ) τ (~λ · ∇u)dx.

We hoosethesepointssothatthedis reteformulaisindependentofthe

num-beringofthemeshpoints. Inourexamples,andfortrianglesandtets,we hoose

theverti esof

T

for

k = 2

;weaddtothesepointsthemidedgepointsfor

k = 3

:

sin ethesepointsaredegreesof freedom,theadditional ost isminimized. In

the aseofquads,wealso hoosetheverti esfor

k = 2

. Theweights

ω

quad used

in(31)are

ω

quad

= 1/#{

quadpoints

}

.

Inthefollowing,wedenote

Ψ

K

σ

= |K|

X

x

quad

ω

quad

"

~λ · ∇ψ

σ



(x

quad

) τ (x

quad

) ~λ · ∇u

h



(x

quad

)

#

.

(32)

Con erning the hoi e of the parameter

θ

K

, we have used in pra ti e the

followingdenition;

θ

K

= 1 − max

σ∈T

"

max

T

∋σ

|u

σ

− ¯

u

K

|

|u

σ

| + |¯

u

K

| + ε

!#

(33)

with

ε

of the order of ma hine zero and

u

¯

K

=

P

σ∈K

u

σ



/n

d

. Typi ally,

θ =

O(h

K

)

in a smooth region and

θ ≡ 1

in a dis ontinuity. The relation (33)

dependsonvaluesof

u

outside

K

,thusitseemsthattheformulaisnot ompa t.

Indeedthisistrue,but fromanalgorithmi pointofview,whatisimportantis

(23)

3.7 Summary of the nal s heme for s alar problems.

The algorithm 1 summarizes the main operations performed for an expli it

implementation,andshowthatthe ompa tnessofthemethodisnotdestroyed.

This aneasilybegeneralizedtoothertypeofiteratives hemes.

Algorithm1Sket h oftheexpli itimplementationof thes hemes. The

eval-uationof

θ

K

( f. equation(33) )iskept ompa tbyupdatingandswappingthe

monitors

θ

σ

and

θ

˜

σ

.

1: Initializeby

θ

σ

= 1

forallDOFs.

2: for Dofor

k = 1

to

k

max

(maximumnumberofiterations)do

3: Set

θ

˜

σ

= 0

forea h

σ

and

Res

σ

= 0

4: for Forea h

K

do 5: evaluatequantities

β

K

σ

,

Φ

K

,

h

K

,

Ψ

K

σ

,and

θ

K

,with

θ

K

= 1 − max

σ∈K

θ

σ

6: evaluate

Φ

K

σ

= β

σ

K

Φ

K

+ θ

K

h

K

Ψ

K

σ

,

(34a) 7: evaluate

ξ

σ

= max(˜

θ

σ

,

|u

σ

− ¯

u

K

|

|u

σ

| + |¯

u

K

| + ε

)

(34b) 8: set

θ

˜

σ

= ξ

σ

, 9: update

Res

σ

= Res

σ

+ Φ

K

σ

.

10: endfor 11: Swap:

θ

σ

= ˜

θ

σ

,

12: Update:

u

n+1

σ

= u

n

σ

− ω

σ

Res

σ

13: endfor

Forany

K

, thetotalresidualis dened by(7) . The splitresiduals

Φ

K

σ

are denedby(13)and

β

K

σ

by(21). Theterm

Ψ

K

σ

isdenedby(32) . Thealgorithm

solves(3)fortheinteriordegreesoffreedomand(4)fortheboundaryones.

Whenthelo alLaxFriedri hss hemeisusedasarstorderbuildingblo k,

asforallthe resultsofthis paper,thes hemeis denoted byLLxFf(for Lo al

LaxFriedri hsltered).

4 Numeri al illustrations for the s alar ase.

We start again with the adve tion problem with initial states and adve tion

speedsdenedby(26)and(27). Theresultsobtainedfor

P

2

interpolationwhen

(24)

left pi ture showsthat, for the dis ontinuous solutionof problem (26), wedo

not get any spurious os illations. The right pi ture instead shows, for

prob-lem(27), thebene e ee t of theextra termin smoothingthe ontours that

now are perfe tly ir ular. We have also run agrid renement study on this

problem using

P

2

and

P

3

approximations. Theresults aresummarized on

ta-ble3. Theleastsquaresslopesobtained onrmtheexpe ted onvergen erates.

Tobettervisualizetheimprovementinthesolutionwhengoingfrom

P

1

to

P

2

spatialinterpolation,we onsider,onthespatial domain

[0, 2] × [0, 1]

,thesolid

body rotationofthe inletprole

u(x) = sin(10πx)

. Inthis asetheadve tion

speedisset to

~λ = (y, 1 − x)

. The ontoursofthenumeri alsolutionsobtained

arereportedongure4. Notethatthe

P

1

runhasbeenperformedonthemesh

obtained by sub-triangulatingthe

P

2

mesh so that exa tly the same number

of DOF is used in the two ases. The dramati improvementbrought by the

P

2

approximationis learlyvisible in the ontourplots, andalso in theoutlet

prolesreportedongure5.

We have also run the linear adve tion test ase (26)-(27) on hybrid mesh

madeoftrianglesandnon orthogonalquadragles.Thedetails ofthe meshes,in

termoftrianglesandquad,aredes ribedin table2.

The gure 6 shows the errors done with se ond, third and fourth order

s hemes. We re overthe expe ted order. Wehave also ompared these errors

withtheresultsobtainedwhenallthequadare utintotwotriangles. Itappears

that the hybrid results are a little bit more a urate. This is not however a

deniteadvantage.

Wetestfurtherthedenitionofthesmoothnesssensor

θ

T

bysolvingthe2D

Burgers'sproblem

∂u

∂y

+

1

2

∂u

2

∂x

= 0

if

x ∈ [0, 1]

2

u(x, y) = 1.5 − 2x

on

y = 0.

The exa t solution onsists in a fan that merges into a sho k whi h foot is

lo atedat

(x, y) = (3/4, 1/2)

. Morepre isely,theexa tsolutionis

u(x, y) =

if

y ≥ 0.5

−0.5

if

− 2(x − 3/4) + (y − 1/2) ≥ 0

1.5

else

else

max

− 0.5, min



1, 5,

x − 3/4

y − 1/2

!

The results obtained on the mesh of gure 1 are displayed on gure 7. For

thesakeof omparison,wegivethese ondandthirdorderresultsonthesame

mesh(hen ethe

P

2

resultshavemoredegreesoffreedom). Therearenospurious

(25)

Themethodalsoworksonmore omplexproblemssu hastheGu kenheimer

Riemann problem. This is a well-known non- onvex onservation law. We

providethis examplefor tworeasons: thesolutionstru ture ismore omplex,

inparti ularafanisendedbyasho k. These ondreasonisthat, eventhough

wedonothaveanyanalyti alproofoftheentropystabilityofthes heme, the

numeri alresultsseemto indi atethattheentropy onditionisproperlymet.

Theproblemisoriginallytimedependent,anddes ribedby

∂u

∂t

+

1

2

∂u

2

∂x

+

1

3

∂u

3

∂y

= 0

u(x, y, 0) =

0

if

0 < arctan

y

x



<

4

1

if

4

< arctan

y

x



<

2

−1

if

2

< arctan

y

x



< 2π

(35)

Thesolutionisselfsimilar,andit anbere astas

u(x, y, t) = v(

x

t

,

y

t

) = v(ξ, ν) ,

wherethefun tion

v

satises

−ξv

ξ

− νv

ν

+

1

2

∂v

2

∂ξ

+

1

3

∂v

3

∂ν

= 0

(36a)

withtheboundary onditions

lim

r→+∞

v(r cos θ, r sin θ) = u(cos θ, sin θ, 0).

(36b)

Solving(36)amountstosolve(35)at

t = 1

. Thisproblemhasbeendis ussedin

[29℄andhasbeendrawntoourattentionbyM.BenArtzi(HebrewUniversity

ofJerusalem). Theux

g(u) =

u

3

3

isnon onvexandthisindu es soni sho ks.

Theexa tsolution onsistsin

ˆ A sho k omingoutfrom theline

y = 0

that movesat thespeed

1/3

in

thepositivedire tion,

ˆ asteadysho kat

x = 0

,

ˆ A sho k omingout from theline

x + y = 0

. Theanalysis of [29℄ using

theself-similarityofthesolutionindi ates thatthelo ationofthissho k

(26)

Tosimulate thisproblem,werewrite(36a)as

∂F (u)

∂ξ

+

∂G(u)

∂ν

+ 2u = 0

(37) with

F (u) =

1

2

u

2

− ξu

and

G(u) =

1

3

u

3

− νu

. Thetotalresidualon

T

takesnow

intoa ountthepresen eofthesour eterm:

Φ

T

=

Z

∂T



F (u)n

x

+ G(u)n

y



dxdy + 2

Z

T

udxdy

Theintegralon

∂T

hasbeenevaluatedbymeansofa3pointGaussian

quadra-tureformula,whilethevolumeintegral anbeeasily omputedexa tly.

Thesolutionisdisplayedon gure8. Wesee thatevenfor thisnon onvex

problem,therearenoos illations losetothedis ontinuities. Moreinterestingly,

the orre tentropysolutionisobtained. Thisisatopi forfurtherinvestigation:

itisknownthattheupwindRDs hemesmayyieldsolutionsthatdonotrespe t

theentropyinequality[30℄. Thisdoesnotseemtobethe asewithour entered

approa h,mostlikelydue tothepresen eoftheterm(32).

5 Extension to systems

Inthisse tion,wedes ribethes hemeforthesystemofthesteadyEuler

equa-tions des ribed by (1a) with the ux (2) and the onserved variables

u

=

(ρ, ρ~u, E)

T

. We assume a perfe t gas equation of state, and

γ = 1.4

in the

appli ations. We denote by

A

(resp.

B

) the Ja obian matrix of the ux

f

1

(resp.

f

2

)withrespe ttothestate

u

.

Thes hemeisadire t extension ofwhat is donein thes alar ase,with a

majormodi ationbe ausethenaturalunknownisave tor,notas alar. We

providethedetailsofthes hemedes riptiononasingleelement

K

sin ethere

isnoambiguity.

All the results obtained in the system ase are given for quadrati

inter-polants. It isof oursepossibletouse ubi (or largerdegree)interpolant,but

sin ewedealwithsteady problems,wedonotexpe tmajorimprovementinthe

solution. This statementis ertainly not orre t forunsteady problemswhi h

isthetopi ofa omingpaper.

5.1 The rst order building blo k

Therstorders hemeis onstru tedontheLaxFriedri hss heme,i.e.,forany

degreeoffreedom

σ ∈ K

,

Φ

σ

=

1

n

K

d

I

∂K

f

(u

h

) · ~ndl + α

K

(u

h

− ¯

u

).

(38) Here

n

K

d

isthenumberofdegreesoffreedomin

K

,hen e

n

K

d

= 6

foratriangle

and

n

K

(27)

in the examples. The total residual

H

∂K

f

(u

h

) · ~ndl

is evaluated by Simpson

formula: if

Γ = [a, b]

is anedgeof

K

and

c =

a+b

2

,weset

Z

Γ

f (x)dl ≡

1

6



f (a) + 4f (c) + f (b)



,

whi h amounts, in our ase, to use a quadrati interpolant of

f

in

K

. The

averagestateis

¯

u

=

X

σ∈T

u

σ

n

K

d

,

and

α

K

is largerthanthe spe tral radiusof theux Ja obians atthe degrees

offreedom. Inpra ti e,itissettotwi ethismaximum.

5.2 Controlling the os illations

Inthes alarpart,the ontrolofos illationsisa hievedbylimiting theratios

Φ

σ

. Inthesystem ase,this quantityhasnomeaning. Hen e,weadapt the

pro edurepresentedin[26℄. Usingtheaveragestate

u

¯

,we omputetheaverage

owdire tion,i.e.

~¯n =

~¯u

||~¯

u||

= (n

1

, n

2

).

ThenweevaluatetheJa obianmatrix

K

n

= A(¯

u

)n

1

+ B(¯

u

)n

2

(39)

whi hisdiagonalizablein

R

. Theeigenve torsare

r

p

for

p = 1, · · · , 4

asso iated

totheeigenvalues

λ

1,2

= ~¯

u · ~¯

n = ||~¯

u||, λ

3

= ||~¯

u|| − ¯

c, λ

4

= ||~¯

u|| + ¯

c.

Last,wedenoteby

p

therighteigenve torsofthesystem,i.e. thelinearforms

su hthat anystateve tor

X ∈ R

4

anbede omposedas

X =

4

X

p=1

p

(X)r

p

.

Ourmethodisthenthefollowing:

1. Wede omposetherstorder residuals

Φ

σ

into hara teristi residuals,

forea h

σ, Φ

σ

=

4

X

p=1

p

σ

)r

p

.

We denote the hara teristi residual by

ϕ

p

σ

= ℓ

p

σ

)

, they satisfy the

onservationrelation:

forea h

p ∈ {1, · · · , 4},

X

σ∈K

(28)

2. Forany

p = 1, · · · , 4

,welimitthe hara teristi sub-residualbythesame

pro edureasinthes alar ase:

β

p

σ

:=



ϕ

p

σ

ϕ

p



+

P

σ

T



ϕ

p

σ

ϕ

p



+

3. We onstru tthelimitedresidualby

Φ

σ

:=

4

X

p=1

β

p

σ

ϕ

p

r

p

.

(40)

The property (11) is satised in a suitable norm. Indeed, if

A

0

denotes the

Hessianofthemathemati alentropyevaluatedat

u

¯

,weknowthatwe annd

aset of eigenve tors

r

l

that are orthogonalfor the metri dened by

A

0

. We

re allthisintheannexA. Indeed,ifwedenoteby

( . , . )

A

0

thes alarprodu t

asso iatedto

A

0

. Wehave

ϕ

p

σ

= (r

p

, Φ

σ

)

A

0

sothat

||Φ

σ

||

2

A

0

=

X

p

p

σ

|

2

|; |ϕ

p

|

2

X

p

p

|

2

≤ ||Φ||

2

A

0

(41)

whereof oursewehaveassumedthattheeigenbasis

{r

p

}

isorthonormal.

The matrix

A

0

is not uniform, but we an nevertheless state that if the

onserved state is su h that the density and the pressure are bounded from

aboveandbelow,allthenormsdenedbythe

A

0

u

)

areequivalentandtheLP

propertyisuniformlysatised.

The last stepis to get anexpli it form of su h a basis. As re alledin the

annexA,thestandardeigenve torsoftheEulerequationsaresimpleandgood

andidatesforthatsin eit aneasilybeshownthattheyareorthogonalforthe

quadrati formdenedbytheentropy. Hen e,thisisourpra ti al hoi e. They

are evaluated as the eigenve torsof (39) where

~n

is the normalizedaveraged

velo ity. It the aseofastagnation point,we hoosethe

x

dire tion.

5.3 Spurious mode ltering pro edure.

Asforthes alar ase,thes heme(7b)(40)produ esverygoodresultsin

(29)

the iterative onvergen e is verypoor, and the resultsare at mostrst order

a urate.

Tothesameproblem, weusethesame ure. Weadd to

σ

)

a orre tion ofthetype

h

K

Z

K



(A, B) · ∇ϕ

σ



τ



(A, B) · ∇u



dx

(42)

wherethematrix

τ

is as alingmatrix.

Several hoi eshavebeentested. Thesimplest oneisadiagonals aling,

τ = α

Id

where

α

has the dimension of the inverse of a speed. We an take

α

as the

inverseofthelargestpossibleeigenvalueofthesystem,i.e.

α = (||~u|| + c)

−1

. A

better hoi eseemstobe

τ = h

−1

K

N

wherethe

N

matrixis

N =



X

~

n

normalto

∂K

K

~

n

+



1

= 2



X

~

n

normalto

∂K

|K

~

n

|



1

.

(43)

Inthisrelation,thesummationistheedgesof

K

(i.e. 3fortriangles,and4for

quadrangles)andif

~n

is anys aled inwardve tornormalto theboundary

∂K

of

K

,wehaveset

K

~

n

= (A, B)~n = An

x

+ Bn

y

.

TheJa obianmatrixareevaluatedat theaveragedstate

¯

ρ =

1

n

K

d

X

σ∈K

ρ

σ

,

~u =

1

n

K

d

X

σ∈K

~u

σ

,

p =

¯

1

n

K

d

X

σ∈K

p

σ

It is shown in [12℄ that if the velo ity

~u 6= 0

, the matrix

P

~

n

normalto

∂K

K

~

n

+

is

alwaysinvertible. Toavoidthissituation,weslightlymodifytheeigenvalues

λ

+

appearingintheevaluationof

L

i

by

λ

+

→ f

λ

+

=

λ

+

if

|λ| > ε

(λ + ε)

2

else.

Thisisreminis entofHarten'sentropyx,buttherolehereisdierent. Here

ε

isasmallnumberoftheform

α(||~u|| + c)

with

α ≡ 0.01

inthenumeri al

appli- ations,but thesimulationsdonotseemtobeverysensitiveto thisparameter

4

. Note that we apply this to ea h of the eigenvalues, even though only the

λ = ~u · ~n

i

needsto bemodied.

4

Indeed,one ouldavoidthisbynotingthat,usingagain[12℄whereade ompositionusing

Figure

Figure 1: Convetion problem : Results obtained with sheme (20)(21) for P 2
Figure 2: Two initialisations showing the reation of spurious modes. We show
Figure 3: Rotation problem : Results obtained with the sheme (20)(21)(34)
Figure 4: Rotation of the smooth prole: u in = sin(10x) . T op: limited LLxF
+7

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