An efficient high-order compact scheme for the unsteady compressible Euler and Navier–Stokes equations

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An efficient high-order compact scheme for the unsteady

compressible Euler and Navier–Stokes equations

Alain Lerat

To cite this version:

Alain Lerat.

An efficient high-order compact scheme for the unsteady compressible Euler and

Navier–Stokes equations.

Journal of Computational Physics, Elsevier, 2016, 322, pp.365-386.

An

efficient

high-order

compact

scheme

for

the

unsteady

compressible

Euler

and

Navier–Stokes

equations

A. Lerat

DynFluidLab.,ArtsetMetiersParisTech,151Boulevarddel’Hopital,75013Paris,France

a

b

s

t

r

a

c

t

Keywords:

Compactschemes Highorder

Unsteadycompressibleflows Taylor–GreenVortex

Residual-BasedCompact(RBC)schemesapproximatethe3-DcompressibleEulerequations witha5th- or7th-orderaccuracyona5×5×5-pointstencilandcaptureshockspretty well without correction. For unsteady flows however, they require a costly algebra to extract thetime-derivative occurring at severalplacesinthe scheme.A newhigh-order time formulationhasbeen recently proposed[13] for simplifyingthe RBC schemes and increasingtheirtemporalaccuracy.Thepresentpapergoesmuchfurtherinthisdirection anddeeplyreconsidersthemethod.Anavatarofthe

RBC schemes

ispresentedthatgreatly reducesthecomputingtimeandthememoryrequirementswhilekeepingthesametypeof successfulnumericaldissipation.Twoand three-dimensionallinearstabilityareanalyzed and themethodis extendedtothe 3-Dcompressible Navier–Stokes equations.Thenew compactschemeisvalidatedforseveralunsteadyproblemsintwoandthreedimension.In particular,anaccurateDNSatmoderatecostispresentedfortheevolutionoftheTaylor– GreenVortexatReynolds1600andPrandtl0.71.Theeffectsofthemeshsizeandofthe accuracyorderintheapproximationofEulerandviscoustermsarediscussed.

1. Introduction

Among high-order methods forcomputing compressible flows on structured meshes, compact schemes are attractive becauseoftheirnarrowgrid-stencilthatsignificantlyreducesthetruncationerrorforagivenaccuracy-orderandmakesthe treatmentofdeformingmeshesandboundaryconditionseasier.Compactschemesforcompressibleflowshavebeenmainly developedascenteredapproximationsinspace,notablyintheworksbyLele[1],CockburnandShu[2],Yee[3]andVisbal andGaitonde[4],withnumerical dissipationbased onartificialviscosities, numericalfilters orlimiters. Upwindcompact schemeshavealsobeenproposedbyTolstykh[5]andFuandMa[6].AnotherinterestingoptionistheuseofResidual-Based Compact(RBC)schemes.Inasuchascheme,theconsistentpartandthenumericaldissipationareexpressedonlyinterms ofcompactapproximationsofthecompleteresidual,i.e. thesumofthetermsinthegoverningequationsincludingthetime derivative.ThesecompactapproximationsarededucedfromPadéformulasinwhichthe inverseoperators areeliminated. OnaCartesianmesh,aRBC schemecanapproximateahyperbolicsystemofconservationlawsind-dimensionwitha 5th-or7th-orderaccuracyona 5d_{-point stencilandcaptureshocksprettywell withoutcorrection.Descriptionandanalysisof}

theseschemescan befound in[7–13]. A relatedapproachdevelopedon unstructuredmeshesistheresidual-distribution method of Abgrall, Deconinckand Ricchiuto [14–18] in which the residuals are distributed to the nodes of triangles or tetrahedrons.

A peculiarity ofthe RBC schemesis themultiple occurrenceof thetime derivative in thescheme (

∂

w

/∂

t occursd

+

1 times in d-dimension). Besides, due to compactness, discrete spatial-operators are applied to each time-derivative. In the first applications ofthe RBC schemesto unsteady problems (see [19,10,11,20]), the time formulation was based on the Gearthree-levelmethodwhichis A-stable. Thisapproachisefficientforcomputingcompressibleflowsinsteadyand slow unsteadyregimes.However, itrequires an iterativemethodtoadvance thesolution(dual-time approachorNewton sub-iterations) and its time accuracy is limited to order 2, which is not sufficient for some unsteady applications. This is the reason whyan explicit time-formulation of highaccuracy has beenproposed in[13].Since a directextraction of thetime-derivativerequiresthesolutionofalargelinearalgebraic-system,an approximatespace-factorizationofthetime operator has been done witha correction to preserve rigorously the high order in space. This leads to the solution of simplelinearsystemsthatareblocktridiagonalforRBC schemesspatially-accurateatorder3or5andblockpentadiagonal for the RBC schemeof order7.Then, an ABM time-integration of orderq (explicitcombination ofAdams–Bashforthand Adams–Moulton methods)is usedto advance the solution.The ABM method offers the advantage ofonly requiringtwo computationspertime-stepofthefluxbalanceandofthetime-derivativeextraction,whatevertheorderq.

Inthe presentpaperwe gomuchfurtherinthequestofsimplifyingthealgorithmandreducing theCPU-timeforthe same accuracylevel.We constructan avatarofthe RBC schemesby dissociating thecomputation ofthe numerical dissi-pation fromthat oftheconsistentpart.Doingso,theresidual-basedfeatureispartiallylost,butthenumericaldissipation of thenew schemediscretizes thesame partial derivative operatorasthe RBC scheme andonlyrequires thesolution of ordinary tridiagonal(uptoorder 5)orpentadiagonallinearsystemswithouttheneedforaspacefactorization.The time-integrationusedforthissimpleschemeistheoptimizedlow-storageRungeKuttamethodR K o6 ofBogeyandBailly[21].

The paperisorganized asfollows.Section2presentsthevariousPadécompactapproximationsusedinthispaperand recalls the construction ofthe RBC schemeand thespecific features ofits numerical dissipationfor solvinga hyperbolic systemofconservationlawsintwo-dimension.Italsorecallsthedirectextractionofthetimederivativewiththe factoriza-tion techniqueandtheRBC

−

ABM schemes.Section 3describesthenewcompactschemefortheEulerequationsandits combinationwith R K o6 in whichthenumericaldissipationis takeninto accountatthelast stageonly. Thenewformof numerical dissipationiscompared tothat ofnon-compactupwindschemesofhighorder. Amultidimensionalanalysisof linearstabilityoftheoverallalgorithmispresentedforthevariousspatialordersandprovesstabilityforCFL numbersequal toorslightlygreaterthanone.Section4presentstheextensionofthenewcompactschemeoforder5tothecompressible Navier–Stokesequations.Todiscretizethesecondderivativeswithvariablecoefficientsona5-pointstencil,wedonotapply twicethePadéapproximationofafirstderivative,butusespecificPadéformulas.Theschemeheredescribedisspatially ac-curateatorder5fortheEulerfluxesandatorder4forgeneralviscousfluxes(E5V 4 scheme)andonlyrequiresthesolution oftridiagonallinearsystemswithconstantcoefficients.InSection5,thenewschemeisappliedtoseveraltest-problems.We firstconsidertheadvectionofavortexduringalongtimeinthehorizontalordiagonaldirection.These2-DEulerproblems allow avalidationoftheaccuracyorderof E5

−

R K o6 forseveralCFL numbers andameasureoftheimportantreduction in computingtime over the RBC5

−

ABM4 scheme. Then the3-D scalar problemofthe diagonaladvection ofa spherical

Gaussian-shape iscomputedbythenewschemeinordertoconfirmtheaccuracyorderin3Daswell asthestability do-main. Successfulcomputationsare done up to theCFL theoretical-limitof 2

/

√

3. Thenext problemis a directnumerical simulation of the Taylor–GreenVortex atRe

=

1600 and Pr

=

0

.

71.Several accuracy orders are considered forthe Euler andviscoustermsonaseriesofmeshes.EnstrophyevolutionsarecomparedtoareferencesolutionandtheCPU-costsare presented.The E5V 4

−

R K o6 schemeproducesaveryaccuratesolutionona2563 meshforamoderateCPU-time.Finally, an Eulercomputation ofshock–vortexinteraction allowsa firstassessmentoftheshock-capturingcapabilitiesofthenew compactscheme,stillusedwithoutanykindofcorrection.ConclusionsaredrawnandfutureworkisplannedinSection6. 2. RBCschemes

2.1. Reminderoncompactapproximations

Aderivative

∂

f

/∂

x canbeapproximatedatanyorderonaregular meshxj

=

j

δ

x usingtwo simplediscrete operators, i.e. adifferenceandanaverageoveronemeshinterval:

(δ

f

)

_j+1

2

=

fj+1

−

fj

(

μ

f

)

j+12

=

1

2

(

fj+1

+

fj

)

where j isanintegerorhalfinteger.Forinstance:

(δ

μ

f

)

j

= (

μ

f

)

_j+1 2

− (

μ

f

)

j−12

=

1 2

(

fj+1

−

fj−1

)

(δ

2f

)

j

= (δδ

f

)

j

=

fj+1

−

2 fj

+

fj−1

(δ

4f

)

j

=

fj+2

−

4 fj+1

+

6 fj

−

4 fj−1

+

fj−2

Table 1

Error-coefficientK2poftheapproximationofaderivativeatorder2p=4, 6, 8.

Error coefficient K4 K6 K8 Non-compact −301 1 140 − 1 630 Compact − 1 180 1 2100 − 1 44100



∂

f

∂

x



j

=



I

−

1 6

δ

2

₊

1 30

δ

4

₋

1 140

δ

6



δ

μ

f

δ

x



j

+

O

(δ

x8

)

(1)

where I is the identity operator. This writingalso contains the approximationsof lower orders. Indeed,by successively removing theterms

−

₁₄₀1

δ

6, ₃₀1

δ

4 and

−

1₆

δ

2,werespectivelygetthe approximationsoforder6,4and2on a7-,5- and 3-pointstencil.

Toobtain thesameaccuracy orders ona shorterstencil,we canuse aPadé compactapproximation ofthe derivative, brieflycalledaPadéderivative,thatis:



∂

f

∂

x



j

=



_∇

f

δ

x



j

+

O

(δ

x2p

)

(2) with

∇ =

D−1

(

I

+

a

δ

2

) δ

μ

,

D

=

I

+

b

δ

2

+

c

δ

4

wherea,b andc arescalarparameters.Theaboveformulainvolves5pointsatmost.However,toextractthePadéderivative

∇

f

/δ

x,wehavetosolvethelinearalgebraicsystem:



D

∇

f

δ

x



j

=



(

I

+

a

δ

2

)

δ

μ

f

δ

x



j (3)

ThematrixassociatedtotheD-operatorispentadiagonalingeneral.

ThePadéderivativeissecond-orderaccurateatleast.Itbecomesoffourthorderatleastifa

=

b

−

1₆.Bychoosing

a

=

0

,

b

=

1

6

,

c

=

0 (4)

bothsidesof(3)havea3-pointstencil.

ThePadéderivativeisaccurateatorder6atleastiftheparameterssatisfy

a

=

1

30

+

6c

,

b

=

1

5

+

6c (5)

Bychoosing c

=

0,theD-operatorhasa3-pointstencil,whichleadstoatridiagonallinearsystem. ThePadéderivativeisaccurateatorder8for

a

=

5 42

,

b

=

2 7

,

c

=

1 70 (6)

The shortgrid-stencilofcompactformulasmakes thetreatment ofdeforming meshesandboundary conditionseasier.In addition,itsignificantlyreducestheerror.Indeed,thetruncationerrorofthePadéderivativeoforder2p

=

4

,

6

,

8 is:



∇

f

δ

x

−

∂

f

∂

x



j

=

K2p

δ

x2p



∂

2p+1f

∂

x2p+1



j

+

O

(δ

x2p+2

)

The truncation error of the non-compact formula (1)has the same form, butthe error-coefficients K2p ofthe compact

formula are much smaller than those ofthe non-compact one, especially as the order increases(see Table 1). Thisis a strongargumentinfavorofcompactness.Notethatasimilarresultholdsforsecondderivatives(see[9]).

Todescribe thenumerical dissipation of theRBC schemes andto approximatethe viscous fluxes, we alsoneed Padé approximationsatmidpointsforaderivativeandanaveraging.Theyaredefinedas:

where

Dδ

=

I

+

bδ

δ

2

+

cδ

δ

4

,

Dμ

=

I

+

bμ

δ

2

+

cμ

δ

4

ThemidpointPadéapproximations _δ∇_x and

μ

aresecond-orderaccurateatleast.Theybecomeoffourthorderatleastif

bδ

−

aδ

=

1

24

,

b

μ

₋

_aμ

₌

1

8 (8)

Theyareaccurateatorder6atleastifinaddition

1 24b δ

₋

_cδ

₌

3 640

,

1 8b μ

₋

_cμ

₌

3 128 (9)

ToconstructthenumericaldissipationoftheRBC schemes,midpointapproximationsoforders2,4and6willbesufficient. BesideswewillchooseDμ

=

Dδ_{,i.e. b}δ

₌

_{bμ andc}δ

₌

_{cμ,toobtainagreatsimplificationinthedissipationterm(vanishing} of Dμ andDδ).Thesimplestrelevantsetsofparametersarethenthefollowingones.For2nd-order:

aδ

=

aμ

=

bδ

=

bμ

=

cδ

=

cμ

=

0 (10)

For4th-order,wesatisfy(8)withDμ

=

Dδ _{andforsimplicityaμ}

₌

_cμ

₌

_0,whichgives

aδ

=

1

12

,

a

μ

₌

₀

_,

_bδ

₌

_bμ

₌

1

8

,

c

δ

₌

_cμ

₌

_{0 (11)}

For6th-order,wesatisfy(8)and(9)withDμ

=

Dδ_,whichgives

aδ

=

11 60

,

a μ

₌

1 10

,

b δ

₌

_bμ

₌

9 40

,

c δ

₌

_cμ

₌

3 640 (12)

Now concerningtheapproximationofviscous terms,thechoicecriterion isnolongerDμ

=

Dδ _butcδ

₌

_cμ

₌

_{0 toproduce} tridiagonallinearsystemsonly.For4th-order,wesatisfy(8)withcδ

=

cμ

=

0 andforsimplicityaμ

=

aδ

=

0,whichgives:

aδ

=

aμ

=

0

,

bδ

=

1

24

,

b

μ

₌

1

8

,

c

δ

₌

_cμ

₌

_{0 (13)}

For6th-order,wesatisfy(8)and(9)withcδ

₌

_cμ

₌

_0,whichgives

aδ

=

17 240

,

a μ

₌

1 16

,

b δ

₌

9 80

,

b μ

₌

3 16

,

c δ

₌

_cμ

₌

_{0 (14)}

2.2. RBC spaceapproximationona5

×

5-pointstencil

Considerthehyperbolicsystemofconservationlaws:

∂

w

∂

t

+

∂

f1

∂

x1

+

∂

f2

∂

x2

=

0 (15)

where t is the time, x1 and x2 are the space coordinates, w is the state vector and f1

=

f1

(

w

)

, f2

=

f2

(

w

)

are flux

componentsdependingsmoothlyonw.TheJacobianmatricesofthefluxaredenoted A1

=

d f1

/

dw and A2

=

d f2

/

dw.

System(15)isapproximatedinspaceon auniformCartesianmesh

(

j1

δ

x1

,

j2

δ

x2

)

usingaresidual-based-compact(RBC)

scheme.Suchaschemeisacompactdiscreteformof

r

=

δ

x1 2

∂(

1r

)

∂

x1

+

δ

x2 2

∂(

2r

)

∂

x2 (16)

wherer istheexactresidual:

r

=

∂

w

∂

t

+

∂

f1

∂

x1

+

∂

f2

∂

x2

andtheright-handsideisaresidual-basednumericaldissipationinwhich



1 and



2arematrices,eachdependingonthe

eigensystems oftheJacobian matrices A1 and A2 andon theratio ofthespace steps.Thesematrices aredefinedin the

sequel.Theycontainnotuningparametersorlimiters.

Todescribethespaceapproximationof(16),weintroducethebasicdiscreteoperatorsineachspacedirection:

(δ

1v

)

_j1+1 2,j2

=

vj1+1,j2

−

vj1,j2

(δ

2v

)

j1,j2+12

=

vj1,j2+1

−

vj1,j2

(

μ1

v

)

_j 1+12,j2

=

1 2

(

vj1+1,j2

+

vj1,j2

)

(

μ2

v

)

j1,j2+12

=

1 2

(

vj1,j2+1

+

vj1,j2

)

ReplacingtheexactresidualinEq.(16)bydifferentcompactcenteredapproximationsr

˜

j1,j2,

(

r

˜

1

)

j1+12,j2 and

(

r

˜

2

)

j1,j2+12,

weobtaintheresidual-basedcompactscheme:

˜

rj1,j2

=

1

2

[δ

1

(

1

˜

r1

)

+ δ

2

(

2r

˜

2

)

]

j1,j2 (17)

Inspiteofappearances,thenumericaldissipationin(17)isnotsimplyoforderonebecause

(

r

˜

1

)

and

(

r

˜

2

)

approximatethe

exactresidualr

=

0.Clearly,ifr

˜

j1,j2 approximatesr atorder2p and

(

˜

r1

)

j1+12,j2,

(

r

˜

2

)

j1,j2+12 approximater atorder2p

−

2,

thenthescheme(17)isaccurateatorder2p

−

1 inspaceanddenotedasRBC2p−1.

Wenowrestricttheschemestencilto5

×

5 points.Forconstructingthemainresidualr

˜

j1,j2,wefirstreplacethespace

derivativesintheexactresidualr byPadéapproximations ∇1

δx1 and

∇2

δx2 oforder2p,wheresimilarlyasin2.1:

∇

l

= (

Dl

)

−1

(

I

+

a

δ

l2

) δ

l

μ

l

,

Dl

=

I

+

b

δ

2l

+

c

δ

4

l

,

l

=

1

,

2 (18)

Thenweapplytheoperator D1D2 toallthetermsintheresidualandobtain

˜

rj1,j2

=



D1D2

∂

w

∂

t

+

D2

(

I

+

a

δ

2 1

)

δ

1μ1f1

δ

x1

+

D1

(

I

+

a

δ

22

)

δ

2μ2f2

δ

x2



j1,j2 (19)

For constructing the two residuals in numerical dissipation

(

r

˜

1

)

_j1+1

2,j2 and

(

r

˜

2

)

j1,j2+12, we use Padé approximations at

midpoints. Moreprecisely, for

(

r

˜

1

)

j1+12,j2 we first replace

∂f1

∂x1 by thePadé derivative atmidpoints

∇1

δx1 andweapply the

Padéaverageatmidpoints

μ

1 to ∂∂wt and

∂f2 ∂x2,whichgives



μ

₁

∂

w

∂

t

+

∇

1f1

δ

x1

+

μ

₁

∂

f2

∂

x2

j1+12,j2 with

∇

l

= (

Dlδ

)

−1

(

I

+

aδ

δ

l2

) δ

l

,

μ

l

= (

D μ l

)

− 1

₍

_I

₊

_aμ

_δ

2 l

)

μ

l D_lδ

=

D_lμ

=

I

+

bδ

δ

_l2

+

cδ

δ

_l4

,

l

=

1

,

2

whereaδ_,aμ,bδ _andcδ _{aretheparameters(10)–(12)givingtheorder2p}

₋

_2. Thenwereplace ∂f2

∂x2 byastandardPadéapproximation

(

∇ 2

δx2

)

j1,j2 with

∇

l

= (

Dl

)

−1

(

I

+

a

δ

l2

) δ

l

μ

l

,

Dl

=

I

+

b

δ

l2

+

c

δ

l4

,

l

=

1

,

2 (20)

wherea,b andcaretheparametersgivingtheorder2p

−

2. Finally,weapplytheoperator Dδ

1D2 toallthetermsandobtain

(

r

˜

1

)

_j1+1 2,j2

=

D₂



(

I

+

aμ

δ

2₁

)

μ1

∂

w

∂

t

+ (

I

+

a δ

_δ

2 1

)

δ

1f1

δ

x1



+ (

I

+

aμ

δ

₁2

)(

I

+

a

δ

2₂

)

δ

2μ2μ1f2

δ

x2



j1+12,j2

Notethatthe operator Dδ

l hascompletely disappearedandthus theparametersbδ andcδ play norole.Similarlywe

con-structthesecondresidualindissipation:

(

r

˜

2

)

_j 1,j2+12

=

D₁



(

I

+

aμ

δ

2₂

)

μ2

∂

w

∂

t

+ (

I

+

a δ

_δ

2 2

)

δ

2f2

δ

x2



+ (

I

+

aμ

δ

₂2

)(

I

+

a

δ

2₁

)

δ

1μ1μ2f1

δ

x1



j1,j2+12

Thus,theresidualsindissipationdependonthefiveparametersaδ_,aμ,a_,b_andc_{.TheparametersoftheRBC3-dissipation} areallzero,thoseoftheRBC5-dissipationsatisfy(11)and(4)andthoseoftheRBC7-dissipationsatisfy(12)and(5).

2.3. RBC dissipation

WenowdiscussthedissipationtermoftheRBC scheme:

˜

dj1,j2

=

1

2

[δ

1

(

1

˜

r1

)

+ δ

2

(

2r

˜

2

)

]

j1,j2 (21)

Thematricesinvolvedaredefinedas:

where,owingtohyperbolicityofSystem(15),TAl isaninvertiblematrixhavingtherighteigenvectorsoftheflux Jacobian-matrix Al (l

=

1

,

2)ascolumnvectorsandDiag

(φ

_l(i)

)

denotesadiagonalmatrixwithdiagonalentries:

φ

₁(i)

=

sgn

(

a(₁i)

)

min



1

,

δ

x2

|

a (i) 1

|

δ

x1m

(

A2

)



,

φ

(₂i)

=

sgn

(

a(₂i)

)

min



1

,

δ

x1

|

a (i) 2

|

δ

x2m

(

A1

)



wherea_l(i)isan eigenvalueof Al andm

(

Al

)

=

min

i

|

a

(i)

l

|

,l

=

1

,

2.Eigenvaluesandeigenvectorsof A1 at

(

j1

+

1

2

,

j2

)

andof

A2at

(

j1

,

j2

+

1₂

)

,i.e. oncellfaces,arecomputedusingtheRoeaverage[22].

The matrices



1 and



2 havebeen designedin [23,7] to introduce some kind ofupwinding. Fora one-dimensional

problem,



1 reducestothesignmatrix:

(

1

)

_j 1+12,j2

=

[sgn

(

A1

)

]j1+12,j2

=



TA1Diag

(

sgn

(

a (i) 1

)

T− 1 A1

j1+12,j2 (22)

Foratwo-dimensionalscalarequation:

∂

w

∂

t

+

A1

∂

w

∂

x1

+

A2

∂

w

∂

x2

=

0 (23)

with Al

=

Al

(

w

)

=

0,l

=

1

,

2,



1 and



2 reducesto

(

1

)

j1+12,j2

=



sgn

(

A1

)

min

(

1

,

1

θ

)



j1+12,j2

,

(

2

)

j1,j2+12

=

[ sgn

(

A2

)

min

(

1

, θ )

]j1,j2+12 (24)

where

θ

characterizestheadvectiondirectionwithrespecttothemesh:

θ

=

δ

x1

|

A2

|

δ

x2

|

A1

|

For

θ

=

1, the advection is along a mesh diagonal. For

θ <

1, the advection is closer to the x1-direction than to the

x2-direction.For

θ >

1,itistheopposite.

Thedissipation term(21)hasbeenanalyzedin[11] forunsteadyproblems.Forsteps

δ

x1 and

δ

x2 ofthesameorderof

magnitude,say

O(

h

)

,aTaylorexpansionof(21)foraRBC schemeoforder2p

−

1 gives:

˜

dj1,j2

=(−

1

)

p−1

_κ

∂

∂

x1





1

(δ

x2p1 −1

∂

2p−1f1

∂

x2p₁ −1

+

χ

δ

x1

δ

x 2p−2 2

∂

2p−1f2

∂

x2p₂ −1

)



+

∂

∂

x2





2

(

χ

δ

x2p1 −2

δ

x2

∂

2p−1f1

∂

x2p₁ −1

+ δ

x 2p−1 2

∂

2p−1f2

∂

x2p₂ −1

)



j1,j2

+

O

(

h2p+1

)

(25)

where

κ

>

0 and

χ

aretwoconstantcoefficientsdependingonlyontheparametersindissipation.Notethat(25)doesnot containtimederivativessincetheyhavebeenreplacedbyspacederivativesusingtheexactsystem(15).

FortheRBC3-dissipation:

κ

=

1 24

,

χ

=

12



b

−

1 6



FortheRBC5-dissipation:

κ

=

1 240

,

χ

=

20



b

−

1 5



FortheRBC7-dissipation:

κ

=

1 5600

,

χ

=

280 3



c

−

1 70



Consideringthescalarequation(23),ithasbeenproved[11]thatanecessaryandsufficientconditionforthepartial differ-entialoperatorin(25)bedissipative,foralladvectiondirections

(

A1

,

A2

)

andfor(



1

,



2)oftheform(24),is

χ

=

0,i.e. no

crossedderivativesin(25).Asimilarresulthasalsobeenprovedin3-D[11]. FortheRBC7-dissipation,

χ

=

0 meansc

=

701.Togetherwiththecondition

a

=

1

30

+

6c



_,

_b

₌

1

5

+

6c

ensuringthe6th-orderaccuracyof ∇2 δx2 in

˜

r1and ∇ 1 δx1 inr

˜

2,weobtain: a

=

5 42

,

b 

₌

2 7

,

c 

₌

1 70

that is a 8th-order accuracy on these

∇

-Padé derivatives as forthe

∇

-Padé derivatives in the main residual. A similar situationoccurswithRBC5- andRBC3-dissipation.Generallyspeaking,acorrectdissipationrequires

∇



= ∇

,thatis

a

=

a

,

b

=

b

,

c

=

c

Fortunately,thisaccuracyenhancementofsometermsinthedissipationoperatordoesnotextendtheschemestencil. Finally,thedissipativeRBC schemes ona 5

×

5 stencilaredefinedbythemainresidual(19)andtheresidualsin dissi-pation:

(

r

˜

1

)

_j1+1 2,j2

=

D2



(

I

+

aμ

δ

2₁

)

μ1

∂

w

∂

t

+ (

I

+

a δ

_δ

2 1

)

δ

1f1

δ

x1



+ (

I

+

aμ

δ

₁2

)(

I

+

a

δ

₂2

)

δ

2μ2μ1f2

δ

x2



j1+12,j2 (26)

(

r

˜

2

)

_j1,j2+1 2

=

D1



(

I

+

aμ

δ

2₂

)

μ2

∂

w

∂

t

+ (

I

+

a δ

_δ

2 2

)

δ

2f2

δ

x2



+ (

I

+

aμ

δ

₂2

)(

I

+

a

δ

₁2

)

δ

1μ1μ2f1

δ

x1



j1,j2+12 (27)

withthefollowingparameters:

RBC3

:

a

=

0

,

b

=

1 6

,

c

=

a μ

₌

_aδ

₌

_{0 (28)} RBC5

:

a

=

1 30

,

b

=

1 5

,

c

=

a μ

₌

₀

_,

_aδ

₌

1 12 (29) RBC7

:

a

=

5 42

,

b

=

2 7

,

c

=

1 70

,

a μ

₌

1 10

,

a δ

₌

11 60 (30)

In theseschemes, the dissipation error (of order 2p

−

1) dominates the dispersiveerror (of order 2p), whichis a good featureforrobustness.Spectralpropertiesoftheseschemesaredescribedin[12].

SincetheRBC5 schemeismostly usedinournumericalapplications,wewritedownitsspecificresiduals(omittingthe

subscripts):

˜

r

= (

I

+

1 5

δ

2 1

)(

I

+

1 5

δ

2 2

)

∂

w

∂

t

+ (

I

+

1 5

δ

2 2

)(

I

+

1 30

δ

2 1

)

δ

1

μ

1f1

δ

x1

+ (

I

+

1 5

δ

2 1

)(

I

+

1 30

δ

2 2

)

δ

2

μ

2f2

δ

x2

˜

r1

= (

I

+

1 5

δ

2 2

)



μ1

∂

w

∂

t

+ (

I

+

1 12

δ

2 1

)

δ

1f1

δ

x1



+ (

I

+

1 30

δ

2 2

)

δ

2μ2μ1f2

δ

x2

˜

r2

= (

I

+

1 5

δ

2 1

)



μ2

∂

w

∂

t

+ (

I

+

1 12

δ

2 2

)

δ

2f2

δ

x2



+ (

I

+

1 30

δ

2 1

)

δ

1μ1μ2f1

δ

x1

TheRBC5-dissipationapproximates:

˜

d

=

1 240



δ

x15

∂

∂

x1

(

1

∂

5f1

∂

x15

)

+ δ

x25

∂

∂

x2

(

2

∂

5f2

∂

x25

)



+

O

(δ

x17

, δ

x27

)

2.4. Directextractionofthetime-derivative

A difficulty inherent to the RBC schemesis that the time derivative occurs, through linear discrete operators dueto compactness,not onlyinthemainresidualbutalsointheother tworesiduals (orthreeresiduals in3-D)involvedinthe numericaldissipation.Toovercomethisdifficulty,thefollowingapproachhasbeenproposedin[13].First,themainresidual andthedissipationaresplitintoapartcontainingthetimederivativeandapurely-spatialpart(stillomittingthesubscripts

˜

r0

=

D2

(

I

+

a

δ

12

)

δ

1μ1f1

δ

x1

+

D1

(

I

+

a

δ

22

)

δ

2μ2f2

δ

x2

˜

r0₁

=

D2

(

I

+

aδ

δ

21

)

δ

1f1

δ

x1

+ (

I

+

aμ

δ

₁2

)(

I

+

a

δ

₂2

)

δ

2μ2μ1f2

δ

x2

˜

r0₂

=

D1

(

I

+

aδ

δ

22

)

δ

2f2

δ

x2

+ (

I

+

aμ

δ

₂2

)(

I

+

a

δ

₁2

)

δ

1μ1μ2f1

δ

x1

Sincethediscreteoperators

δ

1,

δ

2,

μ

1 and

μ

2commutetwobytwo,wecanalsowrited as

˜

˜

d

= (

M1D2

+

M2D1

)

∂

w

∂

t

+

1 2

[δ

1

(

1

˜

r 0 1

)

+ δ

2

(

2r

˜

02

)

]

with M1

=

1 2

δ

1





1

(

I

+

aμ

δ

12

)

μ1

.

,

M1

=

1 2

δ

2





2

(

I

+

aμ

δ

22

)

μ2

.

Owing to the matrix-coefficients



1 and



2, thematricesassociated to thediscrete operators M1 and M2 have ablock

structure. Theyareblock-tridiagonalifaμ

=

0,i.e. forRBC3 andRBC5,orblock-pentadiagonalforRBC7.Theirdimensionis

thenumberofmeshpointsin x1- orx2-directionwithablock-sizeequaltothenumberofequationsintheexactsystem

(15).Forconsistency,thesameblock-structureisgiventothematricesassociatedto D1 and D2 withdiagonalblocks.The

matricesassociatedtoD1andD2 areblock-tridiagonalifc

=

0,i.e. forRBC3 andRBC5,orblock-pentadiagonalforRBC7.

TheRBC schemer

˜

= ˜

d canthusberewrittenas

∂

w

∂

t

=

H0 (31)

where

denotesthediscreteoperator:

=

D1D2

−

M1D2

−

M2D1

andH0 isthespace-fluxcontribution:

H0

= −˜

r0

+

1 2

[δ

1

(

1r

˜

0

1

)

+ δ

2

(

2

˜

r02

)

]

(32)

The time derivative can be obtained by solving the linear system (31). However, the matrix associated to

is rather complicated.Tosimplifyit,adimensionalfactorizationof

isattemptedas:

=

1

2

−

c

where

1

=

D1

−

M1

,

2

=

D2

−

M2

,

c

=

M1M2

As D1 and D2,theone-dimensionaloperators

1 and

2 are locally

O(

1

)

,butsince M1

=

O(δ

x1

)

andM2

=

O(δ

x2

)

,the

correctiveoperator

c islocally

O(δ

x1

δ

x2

)

andcannot beneglected.Thereforethelinearsystem(31)isconsideredinthe

form:

1

2

∂

w

∂

t

=

H0

+

c

∂

w

∂

t (33)

andsolvedapproximatelyinafewiterationsas:

1

2

∂

w

∂

t (m+1)

=

H0

+

c

∂

w

∂

t (m)

,

m

=

0

,

1

, ... ,

mf (34) startingfrom

∂

w

∂

t (0)

=

0

.

(35)

we haveonlyto solvelinearsystemsthatare block-tridiagonalforRBC3 andRBC5 orblock-pentadiagonalforRBC7.It has

beenprovedin[13]thatoneiteration(mf

=

1)forRBC3,twoiterationsforRBC5andthreeiterationsforRBC7 aresufficient

topreservethespaceaccuracyorder.

Thenumericalsolutionisadvancedbysolvingtheordinarydifferentialequation(ODE):

∂

w

∂

t

=

F

(

w

)

(37)

where F

=

−1_H

0.Clearly, the maincost per time-step comes fromthecalculation of F asthesolution ofthe problem

(36).SoweareinterestedinODEmethodsrequiringfewevaluationsofF pertime-step.Severalexplicitmethodsoforder4 werecomparedin[13],namelytheAdams–Bashforthmethod A B4,theextrapolatedBackwardDifferentialFormulae B D F4,

theclassicalRunge–Kuttamethod R K4 andan explicitcombinationoftheAdams–BashforthandAdams–Moultonmethods

calledABM4.ThelatterwasfoundtobetheleastconsuminginCPU-time.ThisABM4 methodwrites:

⎧

⎨

⎩

˜

wn+1

₌

_wn

₊

t 12

(

23Fn

−

16Fn−1

+

5Fn−2

)

wn+1

=

wn

+

₂₄t

(

9F

˜

n+1

+

19Fn

−

5Fn−1

+

Fn−2

)

(38) wherewn

_{= (}

_wn

j1,j2

)

denotesthenumericalsolutionattimelevelt

n

₌

_n

_t,Fn

₌

_F

₍

_wn

₎

_andF

˜

n+1

₌

_F

₍

_w

_˜

n+1

₎

_.

Themaximaltime-stepforstabilityofABM4islowerthanthatofR K4 butmuchgreaterthanthoseofe B D F4 andA B4.

Ontheotherhand,ABM4 requirestwoevaluationsof F pertimestep(Fn andF

˜

n+1)while R K 4 requiresfour.Finally,the

shortestcomputing-timefortheRBC schemeswasobtainedwithABM4.

3. Anewcompactformulation

In order to simplify the RBC scheme (17), the idea is to compute the time derivative in dissipation from the non-dissipativescheme

˜

r

=

0

Fromthedefinition(19)ofthemainresidual

˜

r,weobtain

∂

w

ˇ

∂

t

= −

∇

1f1

δ

x1

−

∇

2f2

δ

x2 (39)

usingthePadécompactderivativesdefinedby(18).Theresidualsindissipation

˜

r1 andr

˜

2become

ˇ

r1

=(

D2

)

−1r

˜

1

= (

I

+

aμ

δ

21

)

μ1

(

∂

w

ˇ

∂

t

+

∇

2f2

δ

x2

)

+ (

I

+

aδ

δ

2₁

)

δ

1f1

δ

x1

ˇ

r2

=(

D1

)

−1r

˜

2

= (

I

+

aμ

δ

22

)

μ2

(

∂

w

ˇ

∂

t

+

∇

1f1

δ

x1

)

+ (

I

+

aδ

δ

2₂

)

δ

2f2

δ

x2

Since

˜

r1,

˜

r2 approximate 0 at a high order and

(

D2

)

−1

=

I

+

O(δ

x22

)

,

(

D1

)

−1

=

I

+

O(δ

x12

)

, the new residuals r

ˇ

1, r

ˇ

2

approximate0atthesameorderas

˜

r1,r

˜

2.

Using(39)toeliminate

∂

w

ˇ

/∂

t,thenewresidualsindissipationreducetothesimpleexpressions:

ˇ

r1

=(

I

+

aδ

δ

12

)

δ

1f1

δ

x1

− (

I

+

aμ

δ

₁2

)

μ1

∇

1f1

δ

x1

ˇ

r2

=(

I

+

aδ

δ

22

)

δ

2f2

δ

x2

− (

I

+

aμ

δ

₂2

)

μ2

∇

2f2

δ

x2 (40)

andthenewcompactschemetakestheform:

∂

w

∂

t

+

∇

1f1

δ

x1

+

∇

2f2

δ

x2

=

1 2

[δ

1

(

1r

ˇ

1

)

+ δ

2

(

2

ˇ

r2

)

]

(41)

Forthe parametersets (28),(29) and(30),thisscheme remains respectivelyaccurate atorder3,5 and 7,butit ismuch simplertoimplement:

– thescheme (41)only requires thesolution ofalgebraic linearsystems withconstant coefficientsto compute the Padé derivatives

∇

1f1

/δ

x1and

∇

2f2

/δ

x2andthesesystems aretridiagonalforscheme-order3and5orpentadiagonalfororder 7,

withoutanyblock.

– Furthermore,thenewschemeisstablewithsimpler



1and



2matricesthanthoserecalledinSection2.3fortheoriginal

RBC scheme.Allthepresentcomputationswiththenewschemehavebeenrunusing:



l

=

sgn

(

Al

)

=

TAlDiag

[

sgn

(

a (i)

l

)

] (

TAl

)

−

1

_,

_l

₌

₁

_,

_{2 (42)}

The linearstability analysispresented below forthe new scheme with(42) justifies thissimplification for 2-D and3-D scalarproblems.

Althoughthisisnotrecommendedforanefficientprogramming,wenowdevelopthedissipationresiduals(40)inorder topreciselyidentifythenewformofdissipation.Fromthedefinition(18)of

∇

l:

Dl

μ

l

∇

l fl

δ

xl

=

μ

lDl

∇

l fl

δ

xl

=

μ

2 l

(

I

+

a

δ

l2

)

δ

lfl

δ

xl

,

l

=

1

,

2

Usingtherelation

μ

2

l

=

I

+

1 4

δ

2 l,weobtain: Dl

μ

l

∇

l fl

δ

xl

= (

I

+

1 4

δ

2 l

)(

I

+

a

δ

l2

)

δ

lfl

δ

xl

,

l

=

1

,

2

Theresiduals(40)canthusbewrittenas:

ˇ

rl

= (

Dl

)

−1Nl

δ

lfl

δ

xl

,

l

=

1

,

2 where Nl

=

Dl

(

I

+

aδ

δ

l2

)

− (

I

+

aμ

δ

2l

)(

I

+

1 4

δ

2 l

)(

I

+

a

δ

l2

),

Dl

=

I

+

b

δ

l2

+

c

δ

l4

AsimplecalculationreducesNl to

Nl

=

n2

δ

_l2

+

n4

δ

4_l

+

n6

δ

_l6 with n2

=

b

−

a

+

aδ

−

aμ

−

1 4

,

n4

=

b a δ

₋

_{a a}μ

₊

_c

₋

a 4

−

aμ 4

,

n6

=

c a δ

₋

a aμ 4

Fortheparameters(28)ofRBC3: n2

= −

₁₂1

,

n4

=

0

,

n6

=

0

Fortheparameters(29)ofRBC5: n2

=

0

,

n4

=

1201

,

n6

=

0

Fortheparameters(30)ofRBC7: n2

=

0

,

n4

=

0

,

n6

= −

₂₈₀₀1

Inserting

ˇ

rl

= (

I

+

b

δ

l2

+

c

δ

4 l

)

− 1

₍

_n 2

δ

_l2

+

n4

δ

_l4

+

n6

δ

6_l

)

δ

lfl

δ

xl

,

l

=

1

,

2

intheright-handsideof(41)andreplacing

∇

l by(18)intheleft-handside,thenewcompactschemetakestheform:

at order 3:

∂

w

∂

t

+

2



l=1

(

I

+

1 6

δ

2 l

)

−1

δ

l

μ

lfl

δ

xl

= −

1 24 2



l=1

δ

l

δ

xl



sgn

(

Al

)(

I

+

1 6

δ

2 l

)

−1

δ

l3fl



(43) at order 5:

∂

w

∂

t

+

2



l=1

(

I

+

1 5

δ

2 l

)

−1

(

I

+

1 30

δ

2 l

)

δ

l

μ

lfl

δ

xl

=

1 240 2



l=1

δ

l

δ

xl



sgn

(

Al

)(

I

+

1 5

δ

2 l

)

−1

δ

l5fl



(44) at order 7:

∂

w

∂

t

+

2



l=1

(

I

+

2 7

δ

2 l

+

1 70

δ

4 l

)

−1

(

I

+

5 42

δ

2 l

)

δ

l

μ

lfl

δ

xl

= −

1 5600 2



l=1

δ

l

δ

xl



sgn

(

Al

)(

I

+

2 7

δ

2 l

+

1 70

δ

4 l

)

−1

δ

l7fl



(45)

Clearly, thedissipationtermofeachabove schemeexpandsasin(25)with

χ

=

0 andthesamecoefficient

κ

.Ittherefore behaves infirstapproximationasthedissipationofthebasicRBC scheme.Notethatthenewschemeremainscompactin itsmainpart(left-handside),butnotinitsdissipativepart.Inpractice,thiscompacitylossisnottooserious,becauseitis easiertolocallymodifythedissipationthantheconsistentpart(forinstanceneara boundary)andalso,aswe seebelow, thisdissipationisonlyusedoncepertimestep,atthelaststageofaRunge–Kuttamethod.

at order 3:

∂

w

∂

t

+

2



l=1

(

I

−

1 6

δ

2 l

)

δ

l

μ

lfl

δ

xl

= −

1 12 2



l=1

δ

l

δ

xl



|

Al

|δ

l3w



(46) at order 5:

∂

w

∂

t

+

2



l=1

(

I

−

1 6

δ

2 l

+

1 30

δ

4 l

)

δ

l

μ

lfl

δ

xl

=

1 60 2



l=1

δ

l

δ

xl



|

Al

|δ

_l5w



(47) at order 7:

∂

w

∂

t

+

2



l=1

(

I

−

1 6

δ

2 l

+

1 30

δ

4 l

−

1 140

δ

6 l

)

δ

l

μ

lfl

δ

xl

= −

1 280 2



l=1

δ

l

δ

xl



|

Al

|δ

l7w



(48) where

|

Al

| =

TAlDiag

[|(

a (i) l

|] (

TAl

)

− 1

_,

_l

₌

₁

_,

₂

Notethatthedissipationcoefficientoftheupwindschemeisalwayshalfofthelastcoefficientintheleft-handside(see[9]

foraproof).

Comparingcentered approximationsofafirst derivative inSection 2.1,we havementionedthat a compactformulais moreaccuratethananon-compactone ofthesameaccuracyorder.Thisyieldsthattheconsistentpart(left-hand-side)of thenewschemes(43)–(45)produceslessdispersive-errorthantheconsistentpartoftheupwindschemes(46)–(48).Now concerning the dissipation, there is some analogy betweenthe right-hand side of the two types ofhigh-order schemes. For a linear problem(constant matrices A1 and A2), these two scheme-types have the same form of first

differential-approximation,buttheschemes(43)–(45)arelessdissipativebecausetheirdissipationcoefficientsaresmaller,namely2,4 and20timessmalleratorder3,5and7,respectively.Sowe canconcludethatboth dispersiveanddissipativeerrorsare smallerfortheschemes(43)–(45).

Wereturntotheusual form(40)–(41)ofthenewschemesandpresenttheir timeintegration.Duetothesimplicityof theseschemes,themostefficienttime-integrationisnolongerABM4 asinSection 2.4forthebasicRBC-schemes,buthas

beenfound intheclassofRunge–Kuttamethods.Specifically, weusethe low-storageexplicitRunge–Kuttamethod R K o6

proposed by Bogey and Bailly [21]. This six-stage method was constructed by optimizing its dispersion and dissipation properties.Ithasbeenshown[21]to bemorestableandmore accuratethanthestandard low-storagemethodwithfour stages(R K s4).

Foradifferentialequationoftheform(37),the R K o6 methodreads:

w(0)

=

wn

w(k)

=

wn

+

α

k

t F

(

w(k−1)

),

k

=

1

,

2

, ... ,

6 (49)

wn+1

=

w(6)

withthefollowingcoefficients:

α1

=

0

.

117979901657

,

α2

=

0

.

184646966491

,

α3

=

0

.

246623604310

,

α4

=

0

.

331839542736

,

α5

=

1

2

,

α6

=

1

.

Moreprecisely, thetime integrationofthe newschemewillbe doneby takingintoaccount thenumericaldissipation at thelaststageonly.Thisreducesthecomputingtimebutnottheaccuracyorderbecausethenumericaldissipationisofthe orderoftheschemespatial-accuracy(3,5or7).Sowereplace(49)by

w(k)

=

wn

+

α

k

t Fk

(

w(k−1)

),

k

=

1

,

2

, ... ,

6 (50)

Fk

= −

∇

1 f1

δ

x1

−

∇

2f2

δ

x2

+

χ

k 2

[δ

1

(

1

ˇ

r1

)

+ δ

2

(

2r

ˇ

2

)

]

(51)

where

χ

k

=

0 fork

=

1

,

2

,

...,

5.Thecoefficient

χ

6isnormallyequalto1,butitmaybereducedinsmoothflows.

WenowpresentaL2-stabilityanalysisofthemethod(50)–(51)fora2-Dadvectionproblem( f1

=

A1w and f2

=

A2w

where A1 and A2 are scalar constants). The Fouriersymbol (in space) of the Padé derivative

t Al

∇

l

/δ

xl is

ı ˙

AlPl, with

˙

Al

=

t Al

/δ

xl and Pl

=

[

1

−

2a

(

1

−

cos

ξ

l

)

]

sin

ξ

l 1

−

2b

(

1

−

cos

ξ

l

)

+

4c

(

1

−

cos

ξ

l

)

2

,

l

=

1

,

2

where

ξ

l isthereducedwave-numberinthexl-direction.

Concerningthedissipation,theFouriersymbolof 1₂

t

δ

l

(

lr

ˇ

l

)

is

−| ˙

Al

|

Ql,where

Ql

= [

1

−

2aδ

(

1

−

cos

ξ

l

)

](

1

−

cos

ξ

l

)

−

1 2

[

1

−

2a

μ

₍

₁

₋

_cos

_ξ

l

)

]

sin

ξ

lPl

,

l

=

1

,

2

TheFouriersymbolof

t Fk istherefore

t



Fk

= −(

Q(k)

+ ı

P

)

where P

= ˙

A1P1

+ ˙

A2P2 andQ(k)

=

χ

k

(

| ˙

A1

|

Q1

+ | ˙

A2

|

Q2

)

.

Theamplificationfactor g

=

g

(ξ

1

,

ξ

2

)

governingtheFourier-transformevolutionofwn (



wn+1

=

g



wn)satisfies

g(0)

=

1

g(k)

=

1

−

α

k

(

Q(k)

+ ı

P

)

g(k−1)

,

k

=

1

,

2

, ...,

6

g

=

g(6)

Denoting g(rk)

= (

g(k)

)

andg(ık)

= (

g(k)

)

,weobtain:

g_r(0)

=

1

,

g(_ı0)

=

0

gr(k)

=

1

−

α

k

(

Q(k)g(rk−1)

−

P gı(k−1)

),

gı(k)

= −

α

k

(

P g(rk−1)

+

Q(k)gı(k−1)

),

k

=

1

,

2

, ...,

6

|

g

|

2

= (

g(r6)

)

2

+ (

g(ı6)

)

2

Thestability domainofthescheme(50)–(51),thatisthepartofthe

( ˙

A1

,

A

˙

2

)

-planeinwhich

|

g

(ξ

1

,

ξ

2

)

|

≤

1 forall

(ξ

1

,

ξ

2

)

,

is plotted on Fig. 1 forvarious space-orders corresponding to scheme coefficients(28)–(30)and forvarious dissipation-coefficients

χ

6 inthelast R K o6-stage.Thestabilityconditionisoftheform:

t

(

|

A1

|

δ

x1

+

|

A2

|

δ

x2

)

≤

η

(52)

The stability domain increases as the space accuracy increases or asthe dissipation coefficient decreases. With

χ

6

=

1,

the

η

-boundis equalto 1

,

1

.

30 and 1

.

80 for thespace-orders3,5and7,respectively. Fora5th-order accuracyinspace,

η

=

1

.

30

,

1

.

82 and 1

.

98 for

χ

6

=

1

,

0

.

5 and 0

.

2, respectively. Note that in a square mesh (

δ

x1

= δ

x2

= δ

x), a sufficient

conditionforstabilityis

t

δ

x



A2 1

+

A22

≤

η

√

2

Considernowthethree-dimensionalhyperbolicsystem

∂

w

∂

t

+

∂

f1

∂

x1

+

∂

f2

∂

x2

+

∂

f3

∂

x3

=

0 (53) with f3

=

f3

(

w

)

and A3

=

d f3

/

dw.

The extensionofthenewcompactschemeto(53)isreallystraightforward.Thetimeintegration(50)hassimplytobe appliedto Fk

= −

∇

1 f1

δ

x1

−

∇

2f2

δ

x2

−

∇

3f3

δ

x3

+

χ

k 2

[δ

1

(

1

ˇ

r1

)

+ δ

2

(

2r

ˇ

2

)

+ δ

3

(

3r

ˇ

3

)

]

(54)

where

χ

k

=

0 fork

=

1

,

2

,

...

,

5.ThethirdPadéderivative

∇

3f3

/δ

x3 isdefinedsimilarlyasin(18)andthethirdresidualin

dissipationr

ˇ

3 asin(40)withamatrix



3 givenby(42)forl

=

3.

The3-Dlinearstabilityanalysisisquitesimilartothe2-Doneandproducesastabilitydomainwithinaregular octahe-dron inthe

( ˙

A1

,

A

˙

2

,

A

˙

3

)

-space.The boundarypointsofthisoctahedrononthethreeaxisare xl

= ±

η

,l

=

1

,

2

,

3.The3-D