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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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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
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
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
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
1 Introdu tion
Inthere entyears,therehasbeenastrongeorttodeveloprobustandhigher
order(
> 2
)s hemesfor hyperboli equations,su h astheEuler equations,onunstru 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 tion3were 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
4. Allthedis retization stepsdes ribedinse tion3arerevisited anddetails
presented in se tion 5. 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 hletboundaryonditions,
u = g.
(1b)denedontheinowboundary
1
∂Ω
−
= {x ∈ ∂Ω, ~n · ∇
u
f
< 0}.
In(1),theve torofunknown
u
belongstoR
p
,andtheux
f
isf
= (f
1
, . . . , f
d
).
In(1a),
S
isasour etermwhi hhereonlydepends ontheunknownu
.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, andE
isthetotal energy. Intheparti 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
tou
.Hereweassumeaperfe tgasequationofstate,
p = (γ − 1)
E −
1
2
ρ||~u||
2
withγ = 1.4
. 1~
n
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
denoteatessellationofthespatialdomainΩ
. Inthispaperτ
h
isassumedtobe omposedoftrianglesandquadsin2D
2
.Ageneri elementisdenotedby
K
. Denote byn
t
thenumberofelementsofthemesh. Themesh parameterh
denotesthemaximum radiusoftheouter ir lesof theelements
K ∈ τ
h
. Theverti esofthemesharedenotedby
{M
i
}
i=1,...,n
s
. Whenthereisnoambiguity,wedenotetheverti esofanelement
K
by1, . . . , 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
{σ
l
}
l=1....,n
dof
thelist ofdegreesoffreedom. Inthe aseofase ondorder RDs heme, the DOF are the verti es of the mesh, that is :
σ
l
= M
l
, ∀ l
. Toonstru tahigherordera urateRD s heme,therearetwooptions:
1. The ontributiontothedis reteequationofaDOF
σ
l
inageneri elementK
is obtained by using information outsideK
. This option has beenfollowed 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
-thdegreepolynomialsexist. Inthis paper,wewillfo usonthe aseofstandard
P
k
and
Q
k
Lagrange
elementsdenedasfollows:
Quadrati interpolation: theDOFsarethesolutionvaluesintheverti es
andtheedgesmidpoints. Thisyields
3 + 3
pointspertrianglein2Dand4 + 6
pointspertetrahedronin3D. Foraquadrangle,weneedtoaddtheentroid,leadingto
4 + 5
points perelementsin 2D.The3D ase wouldneed
27
DOFsperelement.2
Wehaveobtainedresultsin3D,notreportedinthispaper. Wehavenotyet onsidered
the aseofhybridmeshesin3D,eventhoughourmethodshouldextendwithoutproblemsto
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, theDOF arethe 4verti es,2 DOFperedge, and the entroidof ea h fa e,
i.e.
4 + 6 × 2 + 4 = 20
DOFs. The aseofquadrangleelementsleadsto16
DOFperelements,the3D asewouldneed64DOFsperelements.
et .
Notethatthe ontinuityofthestandardLagrangeelementsrequiresthatallthe
DOFonelementboundariesaresharedbyneighboringelements.
Asa onsequen e,inthetriangular/tet ase,we an ountthetotalnumber
ofDOFin termsof thenumberofverti es,edges,fa es(in3D)andelements,
inthe2D ase,wehave
Quadrati :
n
s
+ n
edge
DOFs,Cubi :
n
s
+ 2n
edge
+ n
t
DOFs. inthe3D ase,ifin addition
n
f ace
isthenumberoffa es, wehaveQuadrati :
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
andn
t
≈ 2n
s
andin 3D,
n
edge
≈ 7n
s
,n
f ace
≈ 10n
s
andn
t
≈ 6n
s
. On Table1, we havereportedtheasymptoti 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 onservationonstraint 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)where
f
h
(u
h
)
and
S
h
(u
h
)
arehigh order a urateapproximationsof the
ux
f
(u)
and the sour e termS(u)
. Natural hoi es are: the Lagrangeinterpolantof
f
(u)
atthedegreesoffreedomdeningu
h
,orthetrueux
evaluatedfor
u
h
.
2. if
σ
is a DOF lying on the boundary ofΩ
, the equation forσ
has totake into a ount the boundary onditions. Let
Γ
be any edge/fa e oftheinowboundaryof
Ω
. We onsideranumeri aluxF
whi hdependson theboundary ondition
u
−
, theinwardnormal~n
and the lo al stateu
h
. Then we dene boundary residualsΦ
Γ
σ
whi h satisfy the followingonservationrelation 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 existsv
su h thatu
h
→ v
when
h → 0
, thenv
isaweak 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 fX
p=1
ω
pf
h
(u
h
(x
p))·~n
e
−|K|
GvX
p=1
ω
pS
h
(u
h
(x
p)) := g
Φ
K
.
(7a)havingdenotedby
e
the generi edge (fa e in3d) ofK
,andfor any
Γ ⊂ ∂Ω
−
,
|Γ|
G fX
p=1
ω
pF (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 vdenote 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.
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
Ω
divf
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
Ω
∇ϕ
h
(x) · f
h
(w
h
) +
Z
∂Ω
ϕ
h
(x)f
h
(w
h
) · ~ndl −
Z
Ω
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 itsP
k
/Q
k
Lagrangeapproximation,
ϕ
h
is theLagrange interpolant of{ϕ(σ)}
σ
,Φ
K,c
σ
andΦ
Γ,c
σ
aretheGalerkinresiduals
Φ
K,c
σ
=
Z
K
ψ
σ
divf
(w
h
)−S(w
h
)
dx
andΦ
Γ,c
σ
=
Z
Γ
ψ
σ
F (w
h
, w
−
, ~n)−f (w
h
)·~n
dx ,
(9) andψ
σ
∈ P
k
(K)
orQ
k
(K)
is theLagrangebasisfun tion relativetotheDOF
σ ∈ K
. Using thefa tthat everywheredivf
(w) − S(w) = 0
,theerror anbe3
C
1
0
(Ω)
isthesetofC
1
easilyde omposedas[21,23℄
E(w
h
, ϕ
h
) = −
Z
Ω
∇ϕ
h
(x) · f
h
(w
h
) − f (w)
(
uxapproximationerror)
−
Z
Ω
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 residualsevaluatedon the
P
k
, Q
k
interpolantw
h
satisfyΦ
K
σ
(w
h
) = O(h
k+d
)
(11a) andΦ
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, thenthe trun ationerrorsatises
|E(w
h
, ϕ
h
)| ≤ C(ϕ, f , w) h
k+1
.
The onstant
C(ϕ, f , w)
depends onlyonϕ
,f
,andw
.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 thatΦ
K
σ
= β
K
σ
Φ
K
(12a)thenthe ondition(11)isfullledprovidedthat
β
T
σ
isuniformlybounded. Thisgives 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:
In(13a),wemusthave
X
e∈∂T
|e|
GfX
p=1
ω
pf
h
(u
h
(x
p)) · ~n
e
=
I
∂T
f
h
(u
h
) · ~ndl + O(h
k+d
)
(14a) and|T |
GvX
p=1
ω
pS
h
(u
h
(x
p)) =
Z
T
S
h
(u
h
)dx + O(h
k+d
)
(14b) In(13b) ,wemusthavefortheboundary onditionsintegrals
|Γ|
GfX
p=1
ω
pF (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
[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 aseS = 0
. Inthis setting, all RDs 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 thelastexpressiondenes 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
′
and1 − 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 byu
0
σ
thevaluesoftheinitialsolutionintheDOFlo ations. Amore onvenientapproa hforthedesignofthes hemesistorepla e onditions
(17) bylo al onstraints. Todothis, letusdene forea h DOFthefollowing
mediandualareas:
C
σ
K
=
|K|
n
d
andC
σ
=
X
K∋σ
C
σ
K
Obviously, onditions(17)aremetifthefollowinglo alpositivity onditionsare
veried:
c
K
σσ
′
≥ 0 ∀ σ, σ
′
∈ K
and∀ K
andω
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
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 riptL
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 hthatΦ
H
σ
= β
σ
Φ
K
withX
σ∈K
β
σ
H
= 1
(20)Byanalogy,weintrodu etheparameters
x
σ
dened byx
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 itypreserv-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
1. Conservation.
X
σ∈K
β
σ
= 1
andX
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
ton
d
,anweidentifytheDOF
σ
toitsnumberℓ
in[1, . . . , n
d
]
.Let us onsider in
R
n
d
n
d
linearly independent pointsS = {A
ℓ
}
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
{λ
ℓ
(M )}
withrespe tto
S
:M =
n
d
X
ℓ=1
λ
ℓ
(M )A
ℓ
orequivalently,forany
O ∈ R
n
d
−−→
OM =
n
d
X
ℓ=1
λ
ℓ
(M )
−−→
OA
ℓ
.
Wehavebydenitionn
d
P
ℓ=1
λ
ℓ
(M ) = 1
. Thus,we aninterpret{x
l
}
and{β
l
}
asthebary entri oordinatesofthepoints
L
andH
su hthatL =
n
d
X
ℓ=1
x
ℓ
A
ℓ
H =
n
d
X
ℓ=1
β
ℓ
A
ℓ
andtheproblembe omestodeneamappingonto
R
n
d
: L 7→ H
su hthat the
onstraints
x
ℓ
β
ℓ
≥ 0
aretrue: theadvantageofthat interpretationisthat theonservationpropertiesareautomati allysatised.
Therearemanysolutiontothat problem,oneparti ularlysimpleoneisan
extensionofthePSIlimiter ofStruijs:
β
ℓ
=
x
+
ℓ
X
ℓ
′
x
+
ℓ
′
,
x
+
= max(x, 0).
(21)Thereisnosingularityintheformulasin e
X
ℓ
′
x
+
ℓ
′
=
X
ℓ
′
x
ℓ
′
−
X
ℓ
′
x
−
ℓ
′
≥ 1.
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. UsingtheP
k
interpolantinT
u
h
=
X
σ∈T
u
σ
ψ
σ
,
thetotalresidual
Φ
T
anbewrittenasΦ
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,
sothatΦ
T
=
X
σ∈T
u
σ
k
σ
.
(23) Wenote thatP
σ∈T
k
σ
= 0
. To onstru tanonlinears heme, westart from thefollowingrstorder (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
andc
T
σσ
′
≥ 0
ifα
T
≥ max
σ∈T
|k
σ
|.
(25)This rst order s heme is extremely dissipative, but this is the one from
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
andu(x, y) =
1
ifx = 0
andy > 0
0
ify = 0
andx > 0
(26)These ond problemisobtainedbysetting
~λ = (y, −x)
T
andu(x, y) =
ϕ
0
(x)
ify = 0
0
otherwise (27) whereϕ
0
(x) =
cos
2
(2πx)
ifx ∈ [0.25, 0.75]
0
elseThe 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
Ω
. AssumingageneralLPs heme,weupdatethesolutionby
u
n+1
σ
= u
n
σ
− ω
X
K,σ∈K
β
σ
K
Φ
K
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 xσ
= (
i
N
, 0)
andu
σ
= 0
isσ
is anymid point.Theinitial onditionisdened by
eitherasongure2-a: wepropagate theboundary onditionalongthe
hara teristi softhePDE.
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
]
. WegetΦ
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 thatΦ
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) withΦ
H
σ
denedby(20)and learly,X
σ∈K
Φ
⋆
σ
= Φ
K
be auseP
σ∈K
∇ψ
σ
= 0
.In fa t, (28) has the good ee t of removing the spurious modes that are
Theproblem ofintegral(28)is thatits exa tevaluation requires theexa t
integration of a polynomial of degree
2(k − 1)
whi h is expensive. A betteranalysis 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 tionϕ
, multiply (3) byϕ(σ)
andaddupall theequationsfor alltheDOF ofthe mesh. Using the onservation
relations,andnegle tingboundary onditions,(3)isequivalentto
Z
Ω
ϕ
h
divf
h
(u
h
)dx +
X
K
q
K
(ϕ
h
, u
h
) = 0
(30a) withq
K
(ϕ
h
, u
h
) =
1
n
d
!
X
σ,σ
′
∈
K
(ϕ(σ) − ϕ(σ
′
))
β
σ
K
Φ
K
− Φ
K,c
σ
(30b) withΦ
K,c
σ
theGalerkinresiduals(9). Themodi ationintrodu edin[5℄amountstoadding term(28)to thequadrati form
q
K
. Theproblem is toknowunderwhi h onditionstheresultings hemeisdissipative,keepstheoriginala ura y,
andpreservesthenonos illatorybehaviorof (30a) .
Themostnaturalwayofpro eedingistorepla e(28)byanapproximation
obtainedbymeansofaquadrature formula:
d
K
(ϕ
h
, u
h
) = |K|
X
x
quadω
quad"
~λ · ∇ϕ
σ
(x
quad) τ (x
quad
)
~λ · ∇u
h
(x
quad)
#
(31) su hthat(ϕ
h
, u
h
) 7→
Z
Ω
ϕ
h
divf
h
(u
h
)dx +
X
K
q
K
(ϕ
h
, u
h
) + θ
K
h
K
d
K
(ϕ
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
(ϕ
h
, u
h
) 7→ q
K
(ϕ
h
, u
h
) + θ
K
h
K
d
K
(ϕ
h
, u
h
)
Here,
h
K
is a the radius of the ir le/sphere ir ums ribed toK
. Thepa-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 eofsmoothvariationsofthesolutions,while
θ
K
≪ 1
a rossdis ontinuities.Con erning a ura y on smooth solutions, whatever quadrature formula
weuse, andthankstotheterm
θ
K
h
K
infrontofd
K
,ifu
h
asmoothenoughfun tion su hthat
~λ · ∇u = 0
,thenone anshow(see[18℄for details)thatθ
K
h
K
d
K
(ϕ
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
(ϕ
h
, u
h
) 7→ d
K
(ϕ
h
, u
h
)
to be positivedenite whenever
~λ · ∇u
> 0
. Inparti ular, following[5, 18℄,wewillrequirethat
ω
quad
> 0
andthat
d
K
(ϕ
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 isenoughfor
k = 1
,3pointsareneededfork = 2
,6fork = 3
andsoon: onehasto be exa ton polynomialsof degree
k − 1
. Inthe 3D ase, onequadraturepointisenoughfor
k = 1
,3fork = 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
fork = 2
;weaddtothesepointsthemidedgepointsfork = 3
:sin ethesepointsaredegreesof freedom,theadditional ost isminimized. In
the aseofquads,wealso hoosetheverti esfor
k = 2
. Theweightsω
quad used
in(31)are
ω
quad
= 1/#{
quadpoints
}
.Inthefollowing,wedenote
Ψ
K
σ
= |K|
X
x
quadω
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 thefollowingdenition;
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 andu
¯
K
=
P
σ∈K
u
σ
/n
d
. Typi ally,θ =
O(h
K
)
in a smooth region andθ ≡ 1
in a dis ontinuity. The relation (33)dependsonvaluesof
u
outsideK
,thusitseemsthattheformulaisnot ompa t.Indeedthisistrue,but fromanalgorithmi pointofview,whatisimportantis
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 tbyupdatingandswappingthemonitors
θ
σ
andθ
˜
σ
.1: Initializeby
θ
σ
= 1
forallDOFs.2: for Dofor
k = 1
tok
max
(maximumnumberofiterations)do3: Set
θ
˜
σ
= 0
forea hσ
andRes
σ
= 0
4: for Forea h
K
do 5: evaluatequantitiesβ
K
σ
,Φ
K
,h
K
,Ψ
K
σ
,andθ
K
,withθ
K
= 1 − max
σ∈K
θ
σ
6: evaluateΦ
K
σ
= β
σ
K
Φ
K
+ θ
K
h
K
Ψ
K
σ
,
(34a) 7: evaluateξ
σ
= max(˜
θ
σ
,
|u
σ
− ¯
u
K
|
|u
σ
| + |¯
u
K
| + ε
)
(34b) 8: setθ
˜
σ
= ξ
σ
, 9: updateRes
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: endforForany
K
, thetotalresidualis dened by(7) . The splitresidualsΦ
K
σ
are denedby(13)andβ
K
σ
by(21). ThetermΨ
K
σ
isdenedby(32) . Thealgorithmsolves(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
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]
,thesolidbody rotationofthe inletprole
u(x) = sin(10πx)
. Inthis asetheadve tionspeedisset to
~λ = (y, 1 − x)
. The ontoursofthenumeri alsolutionsobtainedarereportedongure4. 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
bysolvingthe2DBurgers'sproblem
∂u
∂y
+
1
2
∂u
2
∂x
= 0
ifx ∈ [0, 1]
2
u(x, y) = 1.5 − 2x
ony = 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 tsolutionisu(x, y) =
ify ≥ 0.5
−0.5
if− 2(x − 3/4) + (y − 1/2) ≥ 0
1.5
elseelse
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
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
if0 < arctan
y
x
<
3π
4
1
if3π
4
< arctan
y
x
<
3π
2
−1
if3π
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
A sho k omingoutfrom theline
y = 0
that movesat thespeed1/3
inthepositivedire tion,
asteadysho kat
x = 0
, A sho k omingout from theline
x + y = 0
. Theanalysis of [29℄ usingtheself-similarityofthesolutionindi ates thatthelo ationofthissho k
Tosimulate thisproblem,werewrite(36a)as
∂F (u)
∂ξ
+
∂G(u)
∂ν
+ 2u = 0
(37) withF (u) =
1
2
u
2
− ξu
andG(u) =
1
3
u
3
− νu
. ThetotalresidualonT
takesnowintoa ountthepresen eofthesour eterm:
Φ
T
=
Z
∂T
F (u)n
x
+ G(u)n
y
dxdy + 2
Z
T
udxdy
Theintegralon
∂T
hasbeenevaluatedbymeansofa3pointGaussianquadra-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 theappli ations. We denote by
A
(resp.B
) the Ja obian matrix of the uxf
1
(resp.
f
2
)withrespe ttothestateu
.Thes hemeisadire t extension ofwhat is donein thes alar ase,with a
majormodi ationbe ausethenaturalunknownisave tor,notas alar. We
providethedetailsofthes hemedes riptiononasingleelement
K
sin ethereisnoambiguity.
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) Heren
K
d
isthenumberofdegreesoffreedominK
,hen en
K
d
= 6
foratriangleand
n
K
in the examples. The total residual
H
∂K
f
(u
h
) · ~ndl
is evaluated by Simpsonformula: if
Γ = [a, b]
is anedgeofK
andc =
a+b
2
,wesetZ
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
inK
. Theaveragestateis
¯
u
=
X
σ∈T
u
σ
n
K
d
,
and
α
K
is largerthanthe spe tral radiusof theux Ja obians atthe degreesoffreedom. Inpra ti e,itissettotwi ethismaximum.
5.2 Controlling the os illations
Inthes alarpart,the ontrolofos illationsisa hievedbylimiting theratios
Φ
σ
/Φ
. Inthesystem ase,this quantityhasnomeaning. Hen e,weadapt thepro edurepresentedin[26℄. Usingtheaveragestate
u
¯
,we omputetheaverageowdire 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 torsarer
p
forp = 1, · · · , 4
asso iatedtotheeigenvalues
λ
1,2
= ~¯
u · ~¯
n = ||~¯
u||, λ
3
= ||~¯
u|| − ¯
c, λ
4
= ||~¯
u|| + ¯
c.
Last,wedenoteby
ℓ
p
therighteigenve torsofthesystem,i.e. thelinearformssu hthat anystateve tor
X ∈ R
4
anbede omposedasX =
4
X
p=1
ℓ
p
(X)r
p
.
Ourmethodisthenthefollowing:
1. Wede omposetherstorder residuals
Φ
σ
into hara teristi residuals,forea h
σ, Φ
σ
=
4
X
p=1
ℓ
p
(Φ
σ
)r
p
.
We denote the hara teristi residual by
ϕ
p
σ
= ℓ
p
(Φ
σ
)
, they satisfy theonservationrelation:
forea h
p ∈ {1, · · · , 4},
X
σ∈K
2. Forany
p = 1, · · · , 4
,welimitthe hara teristi sub-residualbythesamepro 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 theHessianofthemathemati alentropyevaluatedat
u
¯
,weknowthatwe anndaset of eigenve tors
r
l
that are orthogonalfor the metri dened byA
0
. Were allthisintheannexA. Indeed,ifwedenoteby
( . , . )
A
0
thes alarprodu tasso iatedto
A
0
. Wehaveϕ
p
σ
= (r
p
, Φ
σ
)
A
0
sothat||Φ
⋆
σ
||
2
A
0
=
X
p
|β
p
σ
|
2
|; |ϕ
p
|
2
≤
X
p
|ϕ
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 theonserved state is su h that the density and the pressure are bounded from
aboveandbelow,allthenormsdenedbythe
A
0
(¯
u
)
areequivalentandtheLPpropertyisuniformlysatised.
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 normalizedaveragedvelo 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
the iterative onvergen e is verypoor, and the resultsare at mostrst order
a urate.
Tothesameproblem, weusethesame ure. Weadd to
(Φ
σ
)
⋆
a orre tion ofthetypeh
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
Idwhere
α
has the dimension of the inverse of a speed. We an takeα
as theinverseofthelargestpossibleeigenvalueofthesystem,i.e.
α = (||~u|| + c)
−1
. A
better hoi eseemstobe
τ = h
−1
K
N
wherethe
N
matrixisN =
X
~
n
normalto∂K
K
~
n
+
−
1
= 2
X
~
n
normalto∂K
|K
~
n
|
−
1
.
(43)Inthisrelation,thesummationistheedgesof
K
(i.e. 3fortriangles,and4forquadrangles)andif
~n
is anys aled inwardve tornormalto theboundary∂K
of
K
,wehavesetK
~
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 matrixP
~
n
normalto∂K
K
~
n
+
isalwaysinvertible. Toavoidthissituation,weslightlymodifytheeigenvalues
λ
+
appearingintheevaluationof
L
i
byλ
+
→ f
λ
+
=
λ
+
if|λ| > ε
(λ + ε)
2
4ε
else.Thisisreminis entofHarten'sentropyx,buttherolehereisdierent. Here
ε
isasmallnumberoftheform
α(||~u|| + c)
withα ≡ 0.01
inthenumeri alappli- 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