An example of high order Residual Distribution scheme using non Lagrange elements

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An example of high order Residual Distribution scheme using non Lagrange elements

Remi Abgrall, Jirka Trefilick

To cite this version:

Remi Abgrall, Jirka Trefilick. An example of high order Residual Distribution scheme using non

Lagrange elements. Journal of Scientific Computing, Springer Verlag, 2009, 45 (1-3), pp.3-25. �inria-

00403691�

Lagrange elements

R. Abgrall

(1,2)

and J. Treik

(1,3)

(1) INRIABordeaux Sud Ouest,33 405 Talene, Frane

(2)Institut Polytenhique de Bordeaux, 33 405 Talene,Frane

(3) CTU Prag,

June 8,2009

Abstrat

Weareinterestedinthenumerialapproximationofnonlinearhyperboliproblems. Thepartiular

lassof shemesweare interestedinare thesoalled ResidualDistributionshemes. Intheirurrent

form, theyrelyonthe Lagrangeinterpolationof thepoint valuesofthe approximatedfuntions. This

interpretationofthedegreesoffreedomaspointvaluesplaysafundamentalroleinthederivationofthe

shemes.ThepurposeofthepresentpaperistoshowthatsomenonLagrangeelementsanalsodothe

job,andmaybebetter. ThisopensthedoortoisogeometrianalysisintheframeworkofRDSshemes.

We are interested in the numerial approximation of linear and non linear hyperboli problems. The

partiularlassofshemesweareinterestedinarethesoalledResidualDistributionshemes. Theyanbe

traedbaktotheearlyworkofP.L.Roe[1℄andNi[2℄,butalsotothestabilizedniteelementshemessuh

astheHughes'SUPGsheme[3,4,5℄. Theirmain harateristisarethefollowing: (i)theyhaveanatural

formulation on unstrutured meshes, (ii) their stenil is the most possible ompat one to reah a given

orderofauray,(iii)theirparallelizationisstraightforward. Thesethreepropertiesaresharedinommon

with the Disontinuous Galerkin sheme, but here, thanks to the onformalnature of the approximation,

thenumberofdegreesoffreedomisreduedbyalargefator,asthisanbeseenontable 1.

2D 3D

Order DG RDS DG RD

2

6n

s

n

s

24n

s

n

s

3

12n

s

4n

s

40n

s

8n

s

4

20n

s

9n

s

80n

s

27n

s

Table1:Numberofdegreesoffreedomforthirdandfourthorderapproximationintheaseofatriangular/tet

mesh. DGstandsforDisontinuousGalerkin,RD forResidualDistribution.

Inpreviouspapers,we,andothers[6,7,8,9,10,11, 12,13℄,haveshownhowto ombinemonotoniity

preservingpropertiesandveryhighauray (

≥ 2

^{)ongeneralonformalmeshes, ornon onformalmeshes}

[14,15℄. Oneof thekeyingredientin theonstrutionisthat thedegreesoffreedomanbeinterpretedas

pointvalues. Thepurposeofthepresentpaperistoshowthat somenonLagrangeelementsanalsodothe

job. Thisopensthedoorto isogeometrianalysis [16℄intheframework ofRDSshemes.

The format of the paper is as follows. In a rst part, we reall what are these Residual Distribution

shemes,andshowtheonstrutionofhighordershemes. Amonotoniitypriniple,orvariationdiminishing

one, plays a key role. In the seond part, we provide examples for salar steady non linear hyperboli

equations. Thethird partdisusstheextensiontotheunsteadyaseforawavemodel. Conlusionfollows.

1.1 Introdution

Letusonsiderthefollowingsalarmodelequation,

div

f (u) = S(x) x ∈ Ω ⊂ R

^d

u = g

^{weaklyontheinowboundary}

Γ

−

(1)

where

Γ

−

= {x ∈ ∂Ω, ∇

u

f (u) · ~n(x) < 0}

^,

~n(x)

^{istheoutwardunitnormalof}

Ω

^at

x

^{. In(1),}

u

^and

g

^belong

to

R

, andtheux

f

^has

d

^{omponents,namely}

f = (f

1

, . . . , f

d

)

^{. Weassumethat}

f

^is

C

^{1 and}

g

^belongsto

L

^∞

(Γ)

^{. Thedisussionwillbedevelopedusingthatsalarmodel,with}

d = 2

^{,howeverextensionstosystems}

andthease

d = 3

^areratherstraightforward.

Weonsider atriangulation of

Ω

^{denoted by}

T

_h^{. The triangles are}

{T

_j

}

j=1,...,ne^{. We denote by}

Ω

h

=

∪

j=1,...,ne

T

j^{. Thevertiesofthemesharedenotedby}

{M

i

}

i=1,nv^{. Besidestheusualregularity}assumptions weneed, wealsomakethestandardassumptionthat ifanelement

T

^{hasapartof anedgeon}

Γ

h

:= ∂Ω

h^,

thisfulledgeisinludedin

Ω

h^.

Ineahelement

T

^,weneedanapproximationofthesolution,say

u

^{h,andweassumethefollowingform}

u

^h_|T

= X

σℓ∈T

u

σℓ

ψ

σℓ

|T

.

⁽²⁾

In(2),thesumisindexedbydegreesoffreedomthatareseenaspointsin

T

^{. AtypialexampleisaLagrange}

interpolant. We will assumethat the funtion

u

^{h is ontinuous aross edges, i.e. the}

ψ

σℓ ^{are ontinuous}

arosstheedgesof

T

_h^{,sothat wewrite}

u

^h

= X

σℓ

u

σℓ

ψ

σℓ

.

Morepreisely, given

k ∈ N

, we assumethat foranyfuntion smooth enough

u ∈ C

^k+1

(Ω)

^{, weandene}

u

^h

= π

h

(u)

^{of this type, suh that if}

u

^{is a polynomial of degree}

k

^{, we have}

u = u

^{h. Then, standard}

approximation results, se for example [17℄, show that in

L

^{p norms, we have}

||u − π

h

(u)|| ≤ C(u)h

^k+1.

ThesepropertiesaretrueforexampleusingLagrangepolynomials,Bezier,splinerepresentationsorNURBS

[18, 19℄. We assumethat degreesof freedom also liveon the boundary of

T

^{, this is true for any of these}

examples. Notethat thisassumptionisonsistentwiththeontinuityassumption.

Thankstothis, wedene,ineahelement

T

^{,thetotalresidual}

Φ

T ^as

Φ

T

= Z

∂T

f

^h

(u

^h

) · ~ndl − Z

T

S(x)dx

⁽³⁾

where

f

^{h is some}approximationof the ux

f

^{. Wepreise the}assumptions on

f

^{h abit latterin the text.}

One this hasbeendone, weonsider split-residuals,

Φ

^T_σ^{, for}

σ ∈ T

^{, so that they satisfythe} onservation property:

X

σ∈T

Φ

^T_σ

= Φ

T

.

⁽⁴⁾

Inorder to handle boundaryonditions, we needto onsider boundary residuals. Let

Γ

^{bean edge of}

sometriangle

T

^whihison

Γ

h^{,weonsidertheboundaryresidual}

Φ

Γ

= Z

Γ

F(u

^h

, u

−

, ~n) − g(x) · ~n

dl

⁽⁵⁾

where

(F(u

^h

, u

−

, ~n)

^{isanumerialupwinduxthatdependsonthetraeof}

u

^hon

Γ

^{,theboundaryondition}

u

−

= g

^,withtheunderstandingthatthenumerialuxvanishesonthenonupwind partsoftheboundary.

Then,weonsidersplit-residuals

Φ

^Γ_σ^,for

σ ∈ Γ

^{,sothat theysatisfythe}onservationproperty:

X

σ∈Γ

Φ

^Γ_σ

= Φ

Γ

.

⁽⁶⁾

Onethishasbeendone,theshemewrites: nd

u

^{hsuhthatforanydegreeoffreedom}

σ

^,

•

^If

σ 6∈ ∂Ω

h^,

Σ(u

^h

) := X

T∋σ

Φ

^T_σ

= 0.

^(7a)

•

^If

σ ∈ ∂Ω

h

Σ(u

^h

) := X

T∋σ

Φ

^T_σ

+ X

Γ⊂∂Ω⁻,Γ∋σ

Φ

^Γ_σ

= 0.

^(7b)

Weansummarize(7a)and(7b)by

Σ(u

^h

) = X

E∋i

Φ

^T_σ

= 0

⁽⁷⁾

where

E

^{standseitherforanytriangle}

T

^oredge

Γ

^thatshares

σ

^.

1.2 Design priniples

1.2.1 Consistenywith (1)

Whatarethedesignpriniplesonthesheme(7)with(4)sothatwehaveaonvergentsheme? Theanswer

tothisproblemhasbeenprovidedin[13℄,andwereproduetheresult.

Proposition1.1. Assumethatthemeshisregular,thattheuxapproximation

f

^h

(u

^h

)

^{isontinuousaross}

edges anddenesaonvergentapproximation (in

L

^{1 ofthe}

C

^{1 ux}

f

^{. Assumethat the residualssatisfy the}

onservation relations (4)and (6). Assumethat the sheme (7)denes aunique

u

^{hsuh that}

1. there existaonstant

C(g)

independentof

h

^suhthat

||u

^h

||

_L²

≤ C(g)

^,

2. there exists

v ∈ L

²

(Ω)

^{suhthat asubsequeneof}

u

^{h onvergesto}

v

ⁱⁿ

L

^2,

then

v

^{isaweak solution of (1)}

Theresultof[13℄was aboutarstorderintimeapproximationof

∂u

∂t +

^div

f (u) = 0

withinitial ondition. The adaptationtothesteady ase(1)with boundaryonditionsand souretermis

straightforward,andusesexatlythesamearguments.

1.2.2 Auray

Again,wereallpreviousresults,see[13℄Thekeyremark istosee thatif onean solve (7)aurately,the

shemeisformally

r

^{orderaurateifthe}split-residualsatisfy

Φ

^T_σ

= O(h

^r+d

), Φ

^Γ_σ

= O(h

^k+d−1

).

Thereasonfollowsfrom asimpleerroranalysis. If

ϕ

^{isaompatlysupported testfuntion,letus denote}

ϕ

^{h itsLagrange} interpolationdened by

ϕ

^h

(σ) = ϕ(σ)

^{. Sayingthat, weassumethat within eah triangle,}

thesetofdegreesoffreedomisunisolvant. Theexamplesofsetion2willmakethatpointlearer. Thenwe

multiplytherelations(7)by

ϕ

^h_σ ^{andadd,thenusingthe}onservationrelationsweobtain

E(u

^h

, ϕ

h

) = X

σ∈Ω

ϕ(σ) X

T∋σ

Φ

^T_σ

+ X

Γ⊂∂Ω⁻,Γ∋σ

Φ

^Γ_σ

!

= Z

Ω

div

f

^h

(u

^h

) − S

^h

(u

^h

)

ϕ

h

(x) dx + X

T⊂Ω

1

#{σ ∈ T } X

σ,σ^′∈T

ϕ(σ) − ϕ(σ

^′

)

Φ

^T_σ

− Φ

^T,c_σ

+ Z

∂Ω

F(u

^h

, u

−

, ~n) − f

^h

(u

^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

(u

^h

) + Z

∂Ω

ϕ

h

(x)f

^h

(u

^h

) · ~ndl + Z

Ω

ϕ

h

(x)S

^h

(u

^h

)dx +

Z

∂Ω

F(u

^h

, u

−

, ~n) − f

^h

(u

^h

) · ~n

ϕ

h

(x)dl

+ X

T⊂Ω

1

#{σ ∈ T } X

σ,σ^′∈T

ϕ(σ) − ϕ(σ

^′

)

Φ

^T_σ

− Φ

^T,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

ϕ

h

= π

h

(ϕ)

^,

Φ

^T,c_σ

=

Z

T

ψ

σ

div

f (u

^h

) − S(u

^h

)

dx, Φ

^Γ,c_σ

= Z

Γ

ψ

σ

F(u

^h

, u

−

, ~n) − f (u

^h

) · ~n

dx

and

ψ

σ

∈ P

^k

(T )

^suhthat

ψ

σ

(σ

^′

) = δ

_σ^σ′^.

Followingagain[13℄, havethefollowingresult:

Proposition 1.2. If the solution

u

^{is smooth enough and the residual, applied to the}

P

_k interpolant of

u

satisfy

Φ

^T_σ

(u

^h

) = O(h

^k+d

)

^(9a)

and

Φ

^Γ_σ

= O(h

^k+d−1

),

^(9b)

if moreoverthe approximation

f

^h

(u

^h

)

^is

k + 1

^{-orderaurate,then the trunationerrorsatises}

|E(u

^h

, ϕ

^h

)| ≤ C(ϕ, f, u) h

^k+1

.

The onstant

C(ϕ, u)

^{depends onlyon}

ϕ

^and

u

^.

Westartbyalemma

Lemma1.3. Forthe steady problem (1),ifthe solution

u

^{issmooth,wehave}

Z

∂T

f

^h

(u

^h

) · ~ndl − Z

T

S(x)dx = O(h

^k+d

)

and

Z

∂T

F(u

^h

, u

−

, ~n) − f

^h

(u

^h

) · ~n

dl = O(h

^k+d−1

)

provided that the approximation

f

^h

(u

^h

)

^is

k + 1

^{th order aurate and the numerial ux}

F

^{is Lipshitz}

ontinuous.

Z

∂T

f

^h

(u

^h

) · ~ndl − Z

T

S(x)dx = Z

∂T

f

^h

(u

^h

) · ~n − f (u)

dl

= O(h

^k+1

) × |∂T | = O(h

^k+d

).

Ontheboundary,wehave

Z

∂T

F(u

^h

, u

−

, ~n) − g(x) · ~n

dl = Z

∂T

F(u

^h

, u

−

, ~n) − g(x) · ~n

dl + Z

∂T

F(u, u

−

, ~n) − g(x) · ~n

dl

= Z

∂T

F(u

^h

, u

−

, ~n) − F (u, u

−

, ~n)

dl

and the result follows beause of the approximation inequality and sine the numerial ux is Lipshitz

ontinuous.

Proof of proposition 1.2. This inequalityisaonsequeneof (8)beausewehave

− Z

Ω

∇ϕ

h

(x) · f

^h

(u

^h

) + Z

∂Ω

ϕ

h

(x)f

^h

(u

^h

) · ~ndl + Z

Ω

ϕ

^h

(x)S

^h

(u

^h

)dx =

− Z

Ω

∇ϕ

h

(x) · f (u) + Z

∂Ω

ϕ

h

(x)f (u) · ~ndl + Z

Ω

ϕ

^h

(x)S

^h

(u)dx

!

+ −

Z

Ω

∇ϕ

_h

(x) ·

f (u) − f

^h

(u

^h

)

+ Z

∂Ω

ϕ

h

(x) f (u) − f

^h

(u

^h

)

· ~ndl + Z

Ω

ϕ

^h

(x) S

^h

(u) − S

^h

(u

^h

) dx

!

(10)

where

u

^h

= π

h

(u)

^{. Fromstandard}interpolationresults[17℄, wehave

|ϕ

^h

| ≤ C

^and

|∇ϕ

^h

| ≤ C

^′,

|f

^h

(u

^h

) − f (u)| ≤ C(u, f)h

^k+1and

|S

^h

(u

^h

) − S(u)| ≤ C(u, S)h

^k+1

.

^{sothat(10)isinnormsmallerthat}

C(u, f, S)h

^k+1

forasuitableonstant

C(u, f, S)

^.

Fromlemma1.3,forany

T

^and

Γ

^,

|Φ

^T,c_σ

| ≤ C(u, f, S )h

^{k+d and}

|Φ

^Γ,c_σ

| ≤ C(u, f, S )h

^{k+d−1 where}

d

^isthe

spaedimension.

Then,Forany

T

^,

| X

σ,σ^′∈T

ϕ(σ) − ϕ(σ

^′

)

Φ

^T_σ

− Φ

^T,c_σ

| ≤ X

σ,σ^′∈T

|ϕ(σ) − ϕ(σ

^′

)|

|Φ

^T_σ

| + |Φ

^T,c_σ

|

|

≤ #

^ofelements

× N × ||∇ϕ||

∞

h × C(ϕ, f, S)h

^k+d

where

N

^{isthenumberofdegreeoffreedomineahelement. Inaregularmeshforaboundeddomain,the}

numberof elementssizes like

h

^{−d sothat in theend, wean nda onstant (againdenoted by}

C

^)whih

dependson

u

^,

f

^,

S

^and

Ω

^suhthat

| X

σ,σ^′∈T

ϕ(σ) − ϕ(σ

^′

)

Φ

^T_σ

− Φ

^T,c_σ

| ≤ C(u, f, S, Ω)h

^k+1

.

Thelastestimation isto bedonefortheboundaryterms. Using theonsistenyofthenumerialux,

wersthave

Z

∂Ω

F(u

^h

, u

−

, ~n) − f

^h

(u

^h

, ~n)

ϕ

h

(x)dl

≤ Z

∂Ω

F(u

^h

, u

−

, ~n) − F(u

^h

, u

^h

, ~n)

ϕ

h

(x)dl

≤ L Z

∂Ω

|u

^h

− u

−

| ≤ C(u, f, ∂Ω)h

^k+1

Similarly,wehave,foranyboundaryedge,

|Φ

^Γ,c_σ

| ≤ C(u, f )h

^{k+d. Iftheboundaryof}

Ω

^{isregular,thenumber}

ofboundaryfaesis oftheorderof

h

^−(d−1).

Thus,weget,using againthesamearguments,

X

Γ⊂∂Ω

X

σ,σ^′∈Γ

ϕ(σ) − ϕ(σ

^′

)

Φ

^Γ_σ

− Φ

^Γ,c_σ

≤ C(u, f, ∂Ω)h

^−d+1

h

^k+d

= C(u, f, ∂Ω)h

^k+1

.

Thisompletestheproof.

Letusonludethisparagraphbytwoimportantremarks.

Remark1.4. Weseethat theproof usestwokey elements:

•

^{The problem (1)issteady,}

•

^{One isable toompute}

u

^{h. This isdone in pratievia aniterative algorithm beause the system (7)}

isingeneralnon linear. Inallthe numerial examples, wewillonsider asimpleJaobi-likeiteration,

u

^k+1_σ

= u

^k_σ

− ω

_σ^k

Σ((u

^h

)

^k

)

⁽¹¹⁾

where

ω

_σ^{k isarelaxationparameterthatanbethoughtastheratioofatimestep(onstraintbyaCFL}

ondition)andanarea. The sequene

(u

^h

)

^{k is} initializedtosomevalue (say

u

^h

= 0

^)andmarhedup

toonvergene. The onvergeneissueofthe sequeneisasubtleone, asitwillbeseen.

Theaurayresultwillbetrue,inpratie,providedthatoneisabletoonstrutaonvergentsequene

((u

^h

)

^k

)

k∈N^{,that is,for any}

ε > 0

^,oneannd

N

ε^suhthat

n ≥ N

ε

,

^then

|Σ((u

^h

)

^k

)| ≤ ε.

The algorithm anbe stoppedprovidedthat

ε = O(h

^k

)

^.

1.2.3 Monotoniity preservation

Inthepreviousversions ofthe RDsheme,thedegreesoffreedom were Lagrangepoints,sothat

u

^h_σ ^isthe

valueof

u

^{h at}

σ

^{. Inthatase,theiterativeshemeisdesignedin suh awaythat forany}

k ∈ N

,

max

σ

|u

^k_σ

| ≤ max

σ

max(||g||

∞

, max

σ

|u

⁰_σ

|).

Indeed,theshemeisdesignedsothatforany

σ

^,

σ^′

max

∈V(σ)

|u

^k_σ

| ≤ max

σ^′∈V(σ)

|u

^k−1_σ

|,

where

V (σ)

^{isthesetofneighborsof}

σ

^,

σ

^{inluded. Notethat inthisase,wearenotasking for}

||(u

^h

)

^k

|| ≤ C

⁽¹²⁾

sine it is well known that the Lagrange interpolation, for degree larger than 2, suers from the Gibbs

phenomena.

Anotherwayofthinkingispreiselyto trytoenforetheonstraint(12)globally. Assumethat wehave

ashemethatwrites:

Φ

^E_σ

= X

σ^′∈T

c

^T_σσ′

(u

σ

− u

σ^′

)

^(13a)

where

E

^{iseitheratriangle(aseofaninternaldegreeoffreedom)oraboundaryedge}

Γ

^{(aseofaboundary}

degreeof freedom),with

forany

σ, σ

^′

, c

^T_σσ′

≥ 0.

^(13b)

|u

^k+1_σ

| ≤ max

σ^′∈Vσ

|u

^k_σ′

|

⁽¹⁴⁾

providedthat

ω

σ

≤ X

E∋σ

X

σ^′∈T

c

σσ^′

!

−1

where

E

^{iseither atriangleoraboundaryedge.}

Ifthebasisfuntions

ψ

σ ^{arepositiveweseethat}

|(u

^h

)

ⁿ⁺¹

≤ max

σ

max(||g||

∞

, max

σ

|u

⁰_σ

|)

⁽¹⁵⁾

Anexampleofsuhasplit-residualisgivenbythefollowingLax-Friedrihlikeresidual: werstapprox-

imate

f (u)

^by

f

^h

(u

^h

) := f (u

^h

).

Φ

^T_σ

= Φ

^T

N

T

+ α

T

(u

σ

− u

^T

)

^(16a)

with

u

^T

= P

σ^′∈T

u

σ^′

N

T

, α

T

≥ max

σ^′∈T

Z

T

????

^(16b)

and

N

T ^{beingthenumberofdegreesoffreedomin}

T

^{. Thisfamilyofsplitresidualsdenesashemethatis}

onlyrstorderaurate.

1.3 Constrution of high order shemes

How anwe onstruta shemethat is both monotoniity preserving andhigh order aurate. Using the

remarkontainedin Lemma1.3,onepossibilityistolook forrealnumbers

β

^E_σ

(u

^h

)

⁽

E

^{triangleorboundary}

edge)suh that

Φ

^E_σ

= β

_σ^E

(u

^h

)Φ

T

,

⁽¹⁷⁾

thatareuniformlybounded. Thisensurethat

Φ

^T_σ

= O(h

^k+d

)

^and

Φ

^Γ_σ

= O(h

^k+d−1

)

^.

The question is to dene the

β

^{s suh that the sheme is both high order aurate and} monotoniity preserving.

Arststepisthefollowing: usingamonotoniitypreservingsheme(thinkoftheLaxFriedrihssheme)

whihresidualsaredenoted by

Φ

^L,T_σ ^{whihsatises(13),weformallywrite}

Φ

^H,T_σ

= Φ

^H,T_σ

Φ

^L,Tσ

Φ

^L,T_σ

= X

σ^′∈T

Φ

^H,T_σ

Φ

^L,Tσ

!

c

^L_σσ′

(uσ − u

σ^′

)

= X

σ^′∈T

c

^H_σσ′

(uσ − u

σ^′

)

with

c

^H_σσ′

=

Φ^H,T_σ

Φ^L,Tσ

c

^L_σσ′^{. Hene,sine}

c

^L_σσ′

≥ 0

^,wehave

c

^H_σσ′

≥ 0

^{providedthat ΦH,Tσ}

Φ^L,Tσ

≥ 0

^{. Setting}

x

σ

= Φ

^L,T_σ

Φ

^{T and}

β

σ

= Φ

^H,T_σ

Φ

^T

,

⁽¹⁸⁾

X

σ∈T

x

σ

= X

σ∈T

β

σ

= 1

^andforany

σ ∈ T, x

σ

β

σ

≥ 0.

⁽¹⁹⁾

The problem is to nd a mapping

(x

σ

)

σ∈T

7→ (β

σ

)

σ∈T ^{that satises the onditions (19). This mapping}

annotbelinearaordingtoGodunov'stheorem.

Anextensivedisussionoftheserelationsisdonein[13℄,inpartiularweprovideageometrialinterpre-

tationoftheserelations. Amongthemanymappingsthat satisfy(19),wehavehosen

β

σ

= x

⁺_σ

P

σ^′∈T

x

⁺_σ′

(20)

whihisalwayswelldenedbeause

P

σ^′∈T

x

⁺_σ′

≥ 1

^.

Unfortunately, as we see in the next setion, the resulting sheme (i.e. (7) with (17) and (20) using

the Lax Friedrihs sheme) is over ompressive. The same problem would our with other rst order

spli=residuals,forexamplethoseonstrutedformstandardrstorderux,see[10℄forsomeexamples. The

fundamental reasonisthatthe limitationisdoneaordingto monotoniitypreservingonstraintsonly, in

ompleteignoraneofwhat isthephysisoftheproblem, i.e. howup-windinghasto betriggered intothe

sheme. Hene,weneedtoaddsomedissipationmehanismwithoutdestroyingtheformalaurayinorder

toorretthatdrawbak. . Onewayofdoingthat istoaddto (17)adissipative term,namely

d

T

(ϕ

^h

, u

^h

) = |T | X

xquad

ω

^quad

"

∇

u

f (u

^h

) · ∇ϕ

σ

(x

^quad

)

∇

u

f (u

^h

) · ∇u

^h

− S

(x

^quad

)

#

(21)

suhthatthequadratiform

(v

^h

, u

^h

) 7→ X

σ

v

^h_σ

X

E∋σ

Φ

^E_σ

+ X

T

θ

T

h

T

d

T

(ϕ

^h

, u

^h

)

is dissipative. Again,

E

^{stands for any element or edge that share}

σ

^{. In (21),}

h

T ^{is a the radius of the}

irumsribedirle/sphere,and

θ

T ^{isaparameterthatisoftheorderof}

0

ⁱⁿdisontinuitiesand

1

^elsewhere.

In(21),

x

quad ^anbeinterpretedasquadraturepointsand

ω

^{quadasweights. Saying,weinterpret(21)asa}

disreteversionof

Z

T

∇

u

f (u)ϕ

σ

·

∇

u

f (u)∇u

^h

− S(x)

dx.

However,in [20℄,wehaveshownthat,atleastforlinearux

f (u) = ~λu

^{,isthataneessaryonditionis}

thatthequadratiform

q

K

(v

h

) := X

xquad

ω

^quad

~λ · ∇v

h

(x

^quad

)

2

is positive denite whenever the polynomial

λ · ∇v

^{h is not identially zero. In the ase of polynomial}

interpolation,weneedonlyonequadraturepoint(and

ω

^quad

= 1

^{),forquadrati}polynomials,weneedthree nonalignedpoints(in pratiethevertiesoftheelement,andwetake

ω

^quad

==

¹₃^{,andso on. Detailsan}

befoundin thisreferene,wewillusethistehniquein thepresentpaper.

Therearemanypossiblehoiesfortheparameter

θ

T^{. Forexample,}

θ

T ^{isagoodhoie,eveninthease}

ofdisontinuoussolutionswherewehaveexperimentallynotiedthatno(visible)spuriousosillationour.

However,thebesthoiewehaveexperimentedis

θ

T

= max

σ∈T

max

T∋σ

max

σ^′∈T

|u

σ^′

− u

T

|

|u

_σ′

| + |u

_T

| + ε

!

(22)

with

ε ≈ 10

^{−10. Here,}

u

T

= ( P

σ∈T

u

σ

)/N

^.