Revisiting the spectral analysis for high-order spectral discontinuous methods

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Eprints ID: 17688

To cite this version: Vanharen, Julien and Puigt, Guillaume and Vasseur, Xavier and

Boussuge, Jean-François and Sagaut, Pierre Revisiting the spectral analysis for

high-order spectral discontinuous methods. (2017) Journal of Computational Physics, vol.

337. pp. 379-402. ISSN 0021-9991

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Revisiting

the

spectral

analysis

for

high-order

spectral

discontinuous

methods

Julien Vanharen

a

,

∗

,

Guillaume Puigt

a

,

Xavier Vasseur

b

,

Jean-François Boussuge

a

,

Pierre Sagaut

c

a_Centre_Européen_de_{Recherche et de}_{Formation Avancée}_{en Calcul Scientiﬁque (CERFACS),}₄₂_{avenue Gaspard}_Coriolis,₃₁₀₅₇_Toulouse

Cedex 01, France

b_{ISAE-SUPAERO,}₁₀_avenue_Edouard_Belin,_BP_{54032, 31055}_Toulouse_{Cedex 4,}_France c_Aix_{Marseille Univ, CNRS, Centrale Marseille, M2P2}_UMR_{7340, 13451}_Marseille,_France

a

b

s

t

r

a

c

t

Keywords:

Space–timespectralanalysis Spectraldiscontinuous FiniteDifference Aeroacoustics Aliasing

MatrixPowerMethod

The spectral analysis is a basic tool to characterise the behaviour of any convection scheme.By nature,thesolutionprojectedontotheFourierbasisenablestoestimatethe dissipationandthedispersionassociatedwiththespatialdiscretisationofthehyperbolic linearproblem. Inthispaper,wewishtorevisitsuchanalysis,focusingattentionontwo keypoints.Thefirstpointconcernstheeffectsoftimeintegrationonthespectralanalysis. Itisshownwithstandardhigh-orderFiniteDifferenceschemesdedicatedtoaeroacoustics thatthetimeintegrationhasaneffectontherequirednumberofpointsperwavelength. The situationdepends onthe choiceofthe coupledschemes (one for timeintegration, one for spacederivativeand one for the filter)and here,the compact schemewith its eighth-orderfilterseemstohaveabetterspectralaccuracythantheconsidered dispersion-relation preservingscheme with its associated filter, especiallyin terms of dissipation. Secondly, suchacoupledspace–timeapproachisappliedtothe newclassofhigh-order spectraldiscontinuousapproaches,focusingespeciallyontheSpectralDifferencemethod. Anewwaytoaddressthespecificspectralbehaviouroftheschemeisintroducedfirstfor wavenumbersin_[0,

π

],followingtheMatrix Powermethod.Forwavenumbersabove

π

, an aliasing phenomenon alwaysoccurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing dependson the polynomial degree and onthe number of time steps. A new way to deﬁne dissipation and dispersionis introducedandappliedtowavenumberslargerthan

π

.Sincethenewcriteriarecoverthe previousresultsforwavenumbersbelow

π

,thenewproposed approachisanextension ofallthepreviousones dealingwithdissipationand dispersionerrors.Atlast,sincethe standard FiniteDifference schemescanserveas referencesolutionfortheircapabilityin aeroacoustics,it isshownthattheSpectralDifferencemethodisas accurateas (oreven moreaccurate)thantheconsideredFiniteDifferenceschemes.

*

Correspondingauthor.

E-mail addresses:julien.vanharen@cerfacs.fr(J. Vanharen),guillaume.puigt@cerfacs.fr(G. Puigt),xavier.vasseur@isae.fr(X. Vasseur),

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1. Introduction

Because ofthecontinuous growthofavailable computationalresources during thelast decade,there wasan increased interest in performing LargeEddy Simulation– LES– to solveindustrial problems.Among theseproblems, aeroacoustics requirestocomputeandtotransportaccuratelypressurewavesaroundcomplexgeometriesandoveralongdistance.Many classes of schemes were proposed to perform LES during the last 30 years, depending on the underlying mathematical frameworkconsideredtodiscretisetheNavier–Stokesequations.

First, inthecontext ofFiniteDifference–FD– formalism,high-order centredschemes forstructuredgrids were built following the Taylor’s expansion technique and their accuracy was compared withthe one of spectral methods [1]. Any (high) orderofaccuracycanbeattainedbutthenumberofdegreesoffreedomtoupdatethesolutionatonepointcanbe large. TwooptimisationsofFDschemeswereintroduced:thecompactformulationofLele [2]that leadstoaspectral-like resolution andthe Dispersion–Relation–Preserving – DRP – technique of Tam and Webb [3] dedicated to aeroacoustics. The compact approach linksseveralderivativeswithunknowns locatedclosely. Bythisway, alinear (implicit)systemof equations links all derivatives with unknowns. For a given accuracy, the stencil of DRP schemes is larger than for the standard FDapproachandtheextraunknownsenable tocontrolthenumericalpropertiesofthescheme: dissipationand dispersion. Bothapproachesbeingcentred,theyare nondissipativeandaﬁlterstabilizesthecomputationsby dissipating wavenumbers.Inthispaper,weconsidercompactandDRPschemesasstandardingredientsforLESanditisassumedthat theywillprovidereferenceresults.

Morerecently,anewgenerationofhigh-ordertechniquesdenotedasspectraldiscontinuous emerged.Followingthe pio-neeringworkofReedandHill[4],theDiscontinuousGalerkin–DG–formulationwasfirstappliedtohyperbolicequations byCockburn,Shuandco-authors[5–7]andopenedmanyyearsofresearchandpapers(see[8]asanexampleofreference book onDGmethod).The ideaistosolveproblemsdefinedintheweakforminsideanymeshcell,withoutrequiringthe solutiontobecontinuousatthemeshinterfaces.Attheinterface,thefluxesarecomputedusingstandardRiemannsolvers, asin FiniteVolume– FV–formalism. Therefore,the FVflux computation enablesthecouplingof theweak problemsin surrounding cells andtheFV fluxesmake informationgoing acrossmesh interfaces.Severalalternative high-order meth-odshavebeenrecentlyintroduced.Followingthestaggered-gridmultidomainspectralmethod[9]forstructuredgrids,Liu, Vinokur andWang[10,11]introducedtheSpectral Difference–SD–methodaimingata simplertoimplementandmore efficientmethodthanthecurrentstateoftheartfortheDGmethod.Theapproachwasthenextendedtomixedelements

[12]. The SDmethod takes benefitof the resolution ofthe strong differential formof the equations,asin FD, but does not assume thatthe solutioniscontinuous onthewholemesh,asinFV.Anotherwaytodefine ahigh-order polynomial reconstruction followsthedefinitionofaveragedquantities,asinFV.WiththeSpectral Volume–SV–approach[13–17], a polynomial reconstructionisdefinedinside anycellusingthe averagedquantitiesover sub-cellsbuiltby subdivisionof the initial mesh elements. As before,several Riemannproblems are solved on meshboundariessince the solution poly-nomialsarenot requiredtobecontinuous atmeshinterfaces.Finally,theFlux Reconstructionmethodintroducedin2007 by Huynh [18] solves thestrongformof theequation. Itcan be seenasa collocatedSpectral Differenceschemebutthe maindifferenceoccursinthedefinitionofthefluxpolynomial:now,aliftingoperator[19–22]isintroducedtoincreasethe polynomial degreeoftheinitialfluxpolynomialbyone.Thisismandatorytorecovertherequiredpolynomialdegreeafter thecomputationofthedivergence(hyperbolic)term.ThemainadvantageofFRmethodisitsabilitytorecoverSD,SVand DGapproachesforthelinearadvectionequation,dependingontheliftingoperator[23].Comparedtothestandardschemes forstructuredgrids,themajoradvantageofDG,SV,SDandFRmethodsliesintheirnaturalabilitytohandleunstructured meshes,whichisaprerequisitetotreatcomplexgeometries.Moreover,suchschemesuseaverycompactstencil,defined lo-callyinsideanymeshcell.Thisisalsoanadvantageintermsofhighperformancecomputingrequiredbymassively-parallel LES.

Whendealingwithaeroacoustics,theﬁrstquestiontoanswerconcernsthespectralaccuracyofthechosenscheme:the key pointconcernstherequirednumberofgridpointsper wavelength.The spectralanalysis[24]consistsofdealingwith thespacederivativeandincomparingthenumericalspatialderivativewiththetheoreticalderivative,afterprojectiononto theFourierbasis.Thisanalysisisperformedonthelinearadvectionequationinaperiodicdomainwithaharmonicinitial solution:

⎧

⎨

⎩

∂

u

∂

t

+

c

∂

u

∂

x

=

∂

u

∂

t

+

c

D(

u

)

=

0 u0

(

x

)

=

u

(

0

,

x

)

=

exp

(

jkx

)

with j2

= −

1

,

(1) wherethefunctionu

(

x

,

t

)

istheunknown,c theconstantadvectivevelocity,k theconstantwavenumberand

D

represents thespatialderivativeoperator.Asaconsequence,Eq.(1)willplayanimportantroleandforsakeofclarity,thenotationfor theunknownswillbekeptunchangedalongthewholedocument.

TheformulationoftheFourierspectralanalysismakestheanalysissimpleforstandard schemesbasedonFinite Differ-enceparadigm. This isduetothe fact thatthe degreesoffreedom arecoupled by thenumerics andnotby the method itself. As introduced by Hu etal.in 1999[25] fortheDG method,the situation ismore complexfor spectral discontin-uous methods.ForDG method,the authorsshow that thedegrees offreedom are coupledby thedeﬁnition ofthe local polynomial–insideanymeshcell.Finally,theFourieranalysiscanbeperformedasforFDapproachbuttheﬁnalequation changes.Insteadofoneequationgivingthecomplex-valuednumericalwavenumber,oneobtains,evenforascalarequation,

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asetoflinearequations.Dependingonthepolynomialdegree p, p

+

1 wavesare solutionsofthelinearsetofequations. In[25],itisthenexplainedthatamongthep

+

1 wavestravellingatdifferentphasespeeds,onemodeisthephysicalmode

asitsfrequencyapproximatestheexact dispersionrelationforarangeofwavenumbers,while theothersaretheparasite modes dueto thescheme.In other words,the solutionisthe superpositionofone physicalmode and p parasitemodes. Suchresultswere alsoobtainedby Zhangetal.forthree differentformulationsoftheDGmethod[26].The samekindof analysiswithp

+

1 waves,onephysicalmodeand p parasitemodes,was alsoproposedfortheSVapproach[27],forthe SDapproach[28]andfortheFRtechnique[20,29,30].Finally,eveniftheoccurrenceofparasitemodesisdemonstrated,the proposedanalysisdoesnotexplaintheroleoftheseparasitemodes.Moreover,evenifitcanbeprovedthattheeigenvalues of the systemare periodic (with a periodof 2

π

), the spectral analysis presented in[20,25–30] shows that the spectral behaviour,playingwiththewavenumberandtheorderp ofthepolynomials,isnolonger2

π

-periodic.Thispointisclearly notinagreementwithmathematicalrequirements.

Inthispaper,we introducea newwayto performtheFourierspectral analysisforpolynomialdiscontinuous methods and we compare the accuracy of the SD technique with two standard centred and stabilized FD schemes, the compact schemeofLele[2]andaDRPschemedevelopedbyBogeyandBailly[31].Theremainderofthispaperunfoldsasfollows.In Sec.2,thespace–timeFourierspectralanalysisisappliedtotheconsideredspatialFDschemescoupledwithalow-storage second-order Runge Kuttascheme. In Sec. 3, a new analysis ofthe spectral Fourierapproach forspectral discontinuous methodsisintroducedandappliedtotheSDmethod.InSec.4,weexplicitlyaddressthecaseofawavenumberlargerthan

π

andwe givetheanalysisintermsofnumberofpointsper wavelengthfortheSDmethod.Before concluding,standard FDschemesandSDschemeareﬁnallycomparedinSec.5.

2. Space–timespectralanalysisforhigh-orderFiniteDifferencemethods

Asanintroduction,webrieﬂyexplainhowthespace–timespectralanalysisisperformed.Thisadvancedmethodisthen appliedtotwoFiniteDifferenceschemes.Theresultsobtainedinthissectionwillnotablyplay animportantrolelaterfor purposeofcomparisonwiththenewresultsshowninSec.3.

2.1. Spaceandtimediscretisations

Letus introduce a one-dimensional domaindecomposed ofelements withuniformlength

x. In 1D, anyconvection equationwithaconstantadvectionvelocityislinearandthekeypointconcernsthedeﬁnitionofthenumericalderivative. Letui bethediscrete unknownatthepointi and ui its derivativeatthesamelocation.The lengthofthecomputational

domainisnot takenintoconsideration:thegoalisto deriveschemeexpressions fora givenpointfarfromtheboundary (inordertohaveaccesstothewholeschemestencil).Fornumericalsimulations,thesystemofequationswillbeclosedby applyingperiodicboundaryconditionsbutthisassumptionisnotmandatory toderiveanalytical expressionsfarfromthe boundary.Inthefollowing,twoschemesfrequentlyappliedtoaeroacousticssimulationsareconsidered.

The first scheme is the sixth-order compact scheme ofLele [2]denoted CS6. This scheme is purely centred and for stabilization, dampinghigh frequency waves ismandatory. Here, the compact filter denoted CF8 is the eighth-order fil-terdesignedbyVisbalandGaitonde[32].The Lele’ssixth-ordercompact schemeCS6approximates thefirst-orderspatial derivativewith: 1 3u i−1

+

ui

+

1 3u i+1

=

14 9 ui+1

−

ui−1 2

x

+

1 9 ui+2

−

ui−2 4

x

,

(2)

whiletheeighth-ordercompactﬁlterCF8designedbyVisbalandGaitondeisdeﬁnedby:

α

fuif−1

+

u f i

+

α

fuif+1

=

4

l=0 bl ui+l

+

ui−l 2

,

(3)

wherethesuperscript f meansfiltered.Thebl coefficientsinEq.(3)aregiveninTable 14.Thefiltercoefficient

α

f isequal

to0

.

47.Thisnumericalsetupisusedby,e.g.,Aikensetal.[33]orLeBrasetal.[34].

The second schemeofinterest is theoptimised fourth-order DRPscheme designedby BogeyandBailly [31],denoted FDo11p.Itusesasymmetricstencilwith11pointsanditsexpressionis:

u_i

=

1

x 5

l=−5 alui+l

.

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Theschemeisstabilizedbyanoptimised sixth-orderﬁlterdenotedSFo11p[35]whichisdeﬁnedonthesamestencil:

u_if

=

ui

−

σ

d 5

l=−5

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Table 1

CFLstabilityfortheRKo6s–SFo11p–FDo11pandRKo6s–CF8–CS6combinations. Runge–Kutta Filter Scheme ν

RKo6s SFo11p FDo11p 2.053740

RKo6s CF8 CS6 1.997980

The optimised coeﬃcientsfortheFDo11p scheme(al)andfortheSFo11pﬁlter(dl)aregiveninTable 15.

σd

inEq.(5)is

generallychosensuchthat

σd

=

1

.

0[35].

The advection equation (1) is now time-marched using a standard explicit time integration procedure based on the Runge–KuttaMethod.Inthewholepaper,thelow-storageDRPsecond-ordersix-stageRunge–Kutta(RKo6s)schemeofBogey andBailly [31]ischosen.Forcompleteness,theRunge–KuttacoeﬃcientsaregiveninTable 16.

The full space andtime integration isasfollows. First,the spatial derivative u_i iscomputed thanks tothe numerical scheme(eitherFDo11porCS6).Then,thestandardRunge–Kuttatimeintegrationisperformedandﬁnally,theﬁlter(either SFo11porCF8)isappliedtotheupdatedsolution.

2.2. Space–timeanalysis

The analysis beingperformed in both space and time, u

(

x

,

t

)

is discretised by un_i

=

u

(

i

x

,

n

t

)

, where the index i

(resp. n)referstospace(resp.time)positionand

t tothetimestep.Thefullydiscretespace–timeschemethenreads:

un_i+1

=

un_i

+

6

l=1

γ

l

−

c

t

D(

un_i

)

l

,

(6)

where

D

isthederivativeoperatorintroducedinEq.(1).

IntroducingtheCourant–Friedrichs–Lewy–CFL–numberdeﬁnedby

ν

=

c

t

/

x andthediscretisednormalmode

un_i

=

exp

(

−

jn

ω

t

+

ji k

x

)

into Eq.(6)and applyingthe ﬁltertransferfunction, theexpression ofthe dispersion relationof thespace–time scheme withﬁlteringis:

exp

(

−

j

ω

t

)

=

F

(

k

x

)

1

+

6

l=1

γ

l

(

−

j

ν

km

x

)

l

.

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Eq. (7) contains two new quantities. km represents the modiﬁcation of the spatial mode k when the spatial scheme is

applied.Moreover,

F (

k

x

)

representsthetransferfunctionassociatedtoeitherCF8orSFo11pﬁlters.Theanalytical expres-sionsfor

F

andtheanalyticalrelationsbetweenkm andk aregiveninAppendix Aforconsistency.

Such a dispersion relationmustbe compared withtheexactone

ω

=

kc. Thecorresponding dimensionlessdispersion relationis

ω

t

=

νk

x.Letusnowintroduce:

G

(

k

x

)

=

F

(

k

x

)

1

+

6

l=1

γ

l

(

−

j

ν

km

x

)

l

,

(8)

ϕ

= −

arg

(

G

) /

ν

∈] −

π

,

π

],

(9)

ρ

= |

G

|.

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For a non dispersivescheme,

ϕ

is equal to

ω

t

/

ν

=

k

x. For a non dissipative scheme,

ρ

is equal to 1.

ρ

>

1 gives ampliﬁcation, whereas

ρ

<

1 gives dissipation. Such a coupled space–time approach has been previously introduced in

[36–38].

Remark.Thequantities

ρ

and

ϕ

aredeﬁnedforoneiterationofthetimeintegrationprocess.

2.3. StabilityanalysisbasedonCFLcriterion

Thespace–timediscretisationisstableunderaCFLcondition.BeforeperformingtheFourieranalysis,themaximumCFL number for stablecomputation is computed. Starting fromEq. (6), the iterativeprocess to ﬁnd un

i isconvergent if

ρ

is

strictly lower than 1.Looking forthelargest timestep tomaintain

ρ

strictlylower than1 forallk

x is an optimisation problemandthestabilitylimits(thatweobtained)aresummarizedinTable 1.

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Fig. 1. Effect of time integration on dispersion for the proposed FD schemes.

Fig. 2. Effect of time integration on dissipation for the proposed FD schemes.

2.4. Spectralanalysis

ThreeCFLnumbers(0

.

1,1

.

0 and1

.

9,respectively)arechoseninagreementwiththestabilityconstraintsinTable 1and bothdispersionanddissipationareestimatedfromEq.(8).NumericalresultsareshowninFig. 1andFig. 2fordispersion anddissipation,respectively.ThesecurvesshowthatthespectralbehaviourchangeswiththeCFLnumberandashighlighted inthenextsection,thischangeinspectralbehaviourhasastrongimpactonthenumberofgridpointsperwavelength.

2.5. Numberofpointsperwavelength(PPW)

The numberofpoints per wavelengthisrepresentative oftheresolution powerofanynumericalscheme.The criteria proposed byBogey andBailly [31] arechosen to measurethemaximumwavenumber properly(resp.accurately), deﬁned withsubscript p (resp.withsubscripta) andcalculatedforboth dispersionanddissipation (superscript

φ

and E

respec-tively).Thefourcriteriaareexpressedasanoptimisationproblem.Forthedispersion,theoptimisationproblemsare: Look for kφp

x solution of: max

kx

|

k

x

−

ϕ

|

5

π

×

10 −4

_,

Look for kφa

kx

|

k

x

−

ϕ

|

5

π

×

10 −5

_.

Similarly,theoptimisationproblemsfordissipationare: Look for kE_p

kx

|

1

−

ρ

|

2

.

5

×

10 −3

_,

Look for kE_a

kx

|

1

−

ρ

|

2

.

5

×

10 −4

_.

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Table 2

Numberofpointsperwavelengthforthespace–timecoupledapproachusingtheRKo6s–SFo11p–FDo11pcombination.

ν λφp/x λφa/x λEp/x λaE/x 0.01 3.93 4.65 4.85 5.76 0.1 3.93 4.68 4.85 5.76 0.5 3.89 5.26 4.85 5.76 0.7 3.90 9.64 4.85 5.76 0.9 4.01 10.75 4.85 5.77 1.0 4.15 11.23 4.85 5.77 1.5 5.95 13.28 4.94 5.80 1.9 7.27 14.50 5.63 6.57 Table 3

Numberofpointsperwavelengthforthespace–timecoupledapproachusingtheRKo6s–CF8–CS6combination.

ν λφp/x λφa/x λEp/x λaE/x 0.01 5.42 7.45 4.21 5.66 0.1 5.42 7.42 4.21 5.66 0.5 5.31 6.86 4.22 5.67 0.7 5.27 6.54 4.22 5.68 0.9 5.28 6.38 4.22 5.69 1.0 5.33 6.39 4.22 5.70 1.5 6.20 11.84 4.61 5.77 1.9 7.38 12.96 5.61 6.66

For thecoupledspace–time analysis, theresultsmaynow depend ontheCFL number

ν

.Theresults obtainedforthe considered schemesaresummarizedinTable 2andTable 3.Itcanbefoundthat,whentheCFLnumber

ν

tendsto 0,the relative errorondispersion anddissipation duetothetime integrationprocedure decreasesandthe(standard)resultsfor the (standard)spatialspectral analysisareobviouslyrecovered.In termsofdispersion,therequirednumberofpointsper wavelengthdoesnotnecessaryincreasewhentheCFLnumber

ν

increasessincethedispersioncurvesarenon-monotonic.In contrast,intermsofdissipation,therequirednumberofpointsperwavelengthincreaseswhentheCFLnumber

ν

increases.

2.6. Summary

Ithasbeenshownthatthetimeintegrationprocedurehasaneffectonthespectralbehaviourofthescheme.Ofcourse, standard resultsare recovered when theCFL numbertends to 0. For theCS6 compact scheme,accounting forthe time integration procedure doesnot really change the required number of points per wavelengthup to

ν

=

1. However, the situationdiffersfortheDRPschemeFDo11psincethenumberofpointsperwavelengthmustbeincreased.BothﬁltersCF8 andSFo11parenotspeciﬁcallysensitivetotheCFLnumber

ν

up to

ν

=

1.Asaconsequence,thisanalysisshowsthat the coupledspace–timeapproachisaprerequisitetodeducethenumberofgridpointsperwavelength.

Moreover, a generalmethodto studythespectral behaviour ofspace–time discretisation hasbeenrecalled. As shown nextinSec.3,suchacoupledapproachismandatorytodescribethespectralbehaviourofspectraldiscontinuoustechniques.

3. Spectralanalysisofthespectraldifferencemethod

Thissection isdevotedtotheFourieranalysisoftheSDmethod.Asbefore,theanalysisisperformedontheadvection equation(1).

3.1. Descriptionofthespectraldifferencemethod

The spectraldifferencemethodsolves inanycellthestrongformoftheequationperdirection, asintheFDapproach. TheconceptoftheSDapproachissimple:ifthesolutionu evolvesasapolynomialofdegree p,thedivergenceoftheﬂux must alsobe apolynomial ofdegree p and thereforetheﬂux polynomialmust bea polynomialofdegree p

+

1.Instead ofdefininganypolynomialbyitsmonomialcoefficients,polynomialsaredefinedby theirvaluesonasetofpointsandby Lagrange interpolation.Weintroduce thesetof p

+

1 pointscalledsolutionpoints denotedas

(

S Pl

)

1≤l≤p+1 andthe p

+

2

points calledﬂuxpoints denotedas

(

F Pl

)

1≤l≤p+2.Let

L

l (respectively

T

l) be theLagrange polynomialsofdegree p (resp.

p

+

1) basedonthesolution(resp.ﬂux)pointindexl. Detailsregardingtheproceduretocomputetheﬂuxdivergenceare giveninAppendix B.

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3.2. MatrixformoftheSDprocedure

Since the polynomialsare defined on a Lagrange basis, any polynomial is definedby its valuesat the control points (either solution points orflux points). We introduce Ui

(

t

)

,the columnvector of size p

+

1 (whose componentsare the

solutionsatsolutionpoints)as:

Ui

(

t

)

=

[ui

(

S Pl

,

t

)

]1lp+1

,

(11)

wheremeansthetransposeoperation.

Theﬁrststepistheextrapolationofthesolutionattheﬂuxpointsusingtheformofpolynomials,leadingtothecolumn vectorVi

(

t

)

ofsize p

+

2 deﬁnedby:

Vi

(

t

)

=

[ui

(

F Pl

,

t

)

]1lp+2

=

⎡

⎣

p

+1 r=1 ui

(

S Pr

,

t

)

L

r

(

F Pl

)

⎤

⎦

1lp+2

.

(12)

SinceaRiemannproblemmustbesolved onthecellboundaries, informationsfromneighbouringcellsi

−

1 andi

+

1 are required.Solutionsatsolutionpoints

Ui

(

t

)

andsolutionsatﬂuxpoints

Vi

(

t

)

includingsolutions ontheborderﬂuxpoints

fromadjacentcellsarenowdeﬁnedascolumnvectorsofsize3

× (

p

+

1

)

andp

+

4,respectively:

Ui

(

t

)

=

⎡

⎣

UUi−i1

(

t

(

)

t

)

Ui+1

(

t

)

⎤

⎦ ,

(13)

Vi

(

t

)

=

⎡

⎣

ui−1

F Pp+2

,

t

Vi

(

t

)

ui+1

(

F P1

,

t

)

⎤

⎦ .

(14)

DeﬁningOm,nasthezeromatrixofdimensionm

×

n andIn theidentitymatrixofsizen,theextrapolationmatrixE ofsize

(

p

+

4

)

× (

3

× (

p

+

1

))

suchthat

Vi

(

t

)

=

E

Ui

(

t

)

readsas:

E

=

⎡

⎢

⎣

L

l

F Pp+2

1lp+1 O1,p+1 O1,p+1 O1,p+1 [

L

l

(

F P1

)

]1lp+1 O1,p+1 O1,p+1 [

L

l

(

F P2

)

]1lp+1 O1,p+1

..

.

..

.

..

.

O1,p+1

L

l

F Pp+2

1lp+1 O1,p+1 O1,p+1 O1,p+1 [

L

l

(

F P1

)

]₁lp+1

⎤

⎥

⎦

.

(15)

The second step consists of computing the ﬂux. Here, the Riemann problem is solved using the (upwind) Godunov scheme:

F

Riemann

(

uL

,

uR

)

=

c

1

+

sign

(

c

)

2 uL

+

1

−

sign

(

c

)

2 uR

,

(16)

where sign

(

c

)

=

c

/

|

c

|

. The computation of the flux atthe flux points can be definedby the matrix F of size

(

p

+

2

)

×

(

p

+

4

)

: F

=

c

⎡

⎢

⎣

1

+

sign

(

c

)

2 1

−

sign

(

c

)

2 O1,p 0 0 Op,1 Op,1 Ip Op,1 Op,1 0 0 O1,p 1

+

sign

(

c

)

2 1

−

sign

(

c

)

2

⎤

⎥

⎦

.

(17)

Forthelaststep,theﬂuxpolynomialisdifferentiatedanditsdivergenceiscomputedatthesolutionpoints.The deriva-tivematrixD ofsize

(

p

+

1

)

× (

p

+

2

)

isthen

D

=

T

_l

(

S Pr

)

1rp+1,

1lp+2.

The overall processfor computingthedivergence termfromthe solutionpoints can now bewritten in thefollowing compactmatrixformas:

M1

=

D F E

,

(18)

(9)

Table 4

CFLstabilityboundsfortheSDmethodwithRunge–Kuttatimeintegration (RKo6s). p ν νˆ 2 0.542304 1.626913 3 0.337879 1.351515 4 0.233186 1.165928 5 0.172017 1.032102

3.3. Matrixformforthespectralanalysis

The generatingpatternfora one-dimensionalproblemisgivenby one cell,so letusintroducethe discretisednormal mode within this periodUi

(

t

)

=

Ui

(

t

)

exp

(

ji k

x

)

such that

Ui

(

t

)

=

L

Ui

(

t

)

.Introducing the complex-valued matrix L of

size3

(

p

+

1

)

× (

p

+

1

)

: L

=

⎡

⎣

exp

(

−

Ij kp+

1x

)

Ip+1 exp

(

j k

x

)

Ip+1

⎤

⎦ ,

(19) itcomeseasily:

∂

Ui

(

t

)

∂

t

=

M

Ui

(

t

) ,

(20) where M

= −

D F E L

= −

M1L. 3.4. Timediscretisation

Eq.(20)isasystemoflineardifferentialequationswhoseintegrationiscarriedoutusingthelow-storagesecond-order six-stageRunge–KuttaschemeofBogeyandBailly[31],asinSec.2.

Ui

(

t

)

isdiscretisedby

Un_i

=

Ui

(

n

t

)

andthediscrete

solutionisadvancedintimeusing

Un_i+1

=

Ip+1

+

6

l=1

γ

l

tlMl

Un_i

=

G

Un_i

.

(21)

G isasquarecomplex-valuedmatrixofsize p

+

1 whichaccountsforbothspaceandtimeintegration.G depends onthe polynomialdegree p,onthewavenumberk andofcourseontheCFLnumber

ν

=

c

t

/

x.Itisimportanttonotethatwe keepherethestandarddeﬁnitionoftheCFLnumberforanadvectionequation.

3.5. StabilityanalysisbasedonCFLcriterion

The SD method withthe Runge–Kutta time integration is stableunder a CFL condition, similarly as the FD method. Starting fromEq.(21),the matrix geometric progression between

U_in+1 and

Un_i isconvergent if thespectral radius

ρ

of matrix G isstrictlylowerthan1.ThestabilitylimitsfoundaresummarizedinTable 4.TobeabletocomparewiththeFD method,letusdeﬁneanewCFLnumberas

ν

= (

c

t

/

x

) (

p

+

1

)

=

ν

(

p

+

1

)

withalengthscalewhichcorrespondstothe meandistancebetweentwoadjacentdegreesoffreedom.

3.6. ApplicationoftheNyquist–Shannonsamplingtheorem

TheNyquist–Shannonsamplingtheoremstatesthatatleasttwopointsaremandatorytocaptureagivenfrequency.Such an approachisroutinelyappliedinstandardschemessuchasFD.Here,letusconsideravector

Un_i sampledoveracellof size

x.Thesamplingfrequency fsistherefore fs

=

1

/

x.ItisassumedthatthevectorUi representsthenormalmodeas

inSec.3.3:

Ui

(

t

,

k

)

=

Ui

(

t

)

exp

(

ji k

x

) .

(22)

Computing now thesamekind ofrelationfora newnormalmode withk

=

k

+

m

(

2

π

/

x

)

=

k

+

m

(

2

π

fs

)

(m

∈ Z

), one

ﬁnds: Ui

t

,

k

=

Ui

(

t

)

exp

ji k

x

,

=

Ui

(

t

)

exp

(

ji

(

k

+

2m

π

/

x

)

x

) ,

=

Ui

(

t

)

exp

(

ji k

x

+

i j 2m

π

) ,

=

Ui

(

t

,

k

) ,

(10)

because im

∈ Z

. As a consequence, any normal mode Ui

(

t

,

k

+

2m

π

/

x

)

with m

∈ Z

has the samerepresentation after

samplingatthesolutionpointsandwenotethat,inordertoavoidaliasing,k

x shouldbelongto[0

,

π

].Thisisakeypoint andinall previous studieson spectral accuracy(forDG, SD,SVandFR),such aproperty was always lost[20,25–30].An illustrationofthealiasingphenomenonfork

x

>

π

willbeshowninSec.4.

3.7. Matrixpowermethodforthespectralanalysis

Eq.(21)introducesthetransfermatrixG betweentimestepsn andn

+

1:

Un_i+1

=

G

Un_i

.

(23)

andusingpropertiesofgeometricprogressionsandEq.(23),oneobtains,raisingG tothepowern:

Un_i

=

Gn

U0_i

.

(24)

Carrying out an unsteadycomputation (for theadvection equation)with the Spectral Differencemethod simply consists of computingthen-th power ofthe matrix G inorder toobtain the solution atthen-th time iteration fromthe initial solution.Thecomputation ofthen-thpowerofanymatrixisatthe coreofthe MatrixPowerMethod (MPM)alsocalled PowerIterationorPowerMethod[39,40].ThemainpropertiesoftheMPMaresummarizedbelowforcompleteness.

TheMPMgivesawell-knownalgorithmforthecomputationofthespectralradiusofamatrix,i.e. theeigenvaluewith the largest modulus. The proof of the theorem can be found in [39] and it is introduced hereafter since it enables to establisha generalresultforthespectral analysisoftheSD Method.R3:SinceEq.(23)istheexpressionofamatrixgeometric progression,itsbehaviourisgivenbytheeigenvaluewiththelargestmoduluswhenntendstoinﬁnitybecauseamongallthemodes, theconsideredonehastheleastdissipation.Thisisthereasonwhy,evenifalltheeigenvalueswerecomputed,onlytheonewiththe largestmoduluswasretainedaccordingtotheMPM.

3.7.1. ThematrixpowermethodforSD

G is a

(

p

+

1

)

× (

p

+

1

)

matrix withcomplexcoeﬃcients. Considerthe eigenvalueproblemG

Ui

= λ

Ui,where

Ui

=

0,

Ui

∈ C

(p+1)×1and

λ

∈ C

.

Uiisindeedacolumnvector.

Assumptions.We assume that the eigenvalue problemGvl

= λ

lvl with

λ

l

∈ C

andvl

∈ C

(p+1)×1 admits a complete

nor-malizedeigenvectorspace

v1

,

v2

, ...,

vp+1

spanning

C

(p+1) _with_{corresponding}_eigenvalues_satisfying

_|λ

1

| < |λ

l

|

,l

=

1 and

|λ

l

| <

λ

p+1

,l

=

p

+

1.

Fromnowon,lettheinitialcondition

U0

i beagivenvector,forwhich

U0_i

=

p+1

l=1

α

_l(0)vl

,

with

α

(p0+1)

=

0

,

(25) andlet

∀

n

∈ {

1

,

2

, ...

} ,

U_in

=

G

Un_i−1

,

(26)

bethebasicrecursivesequence.FromEq.(25)andEq.(26),itcomesimmediately:

Un_i

=

p+1

l=1

α

_l(0)Gnvl

.

Furthermore,Gn_v

l

= λ

nlvl,foralll

∈ {

1

,

2

, ...,

p

+

1

}

,andasaconsequence:

Un_i

=

p+1

l=1

α

_l(0)

λ

n_lvl orequivalently:

Un_i

=

α

(_p0₊)₁

λ

n_p₊₁

vp+1

+

p

l=1

α

_l(0)

α

(_p0₊)₁

λ

l

λ

p+1

n

vl

.

Hence,using

λl λp+1

=

1

−

λp+1−|λl| λp+1 ,itcomes:

(11)

Fig. 3. Spectral analysis for the RKo6s–SD schemes with CFL conditionν=0.1: effect of the order of the solution reconstruction.

vp+1

−

1

α

(_p0₊)₁

λ

n_p₊₁

Un_i

∞

p

l=1

α

_l(0)

α

(_p0₊)₁

1

−

ap+1

n

,

(27) where ap+1

=

min l∈{1,...,p} l=p+1

λ

p+1

−|

λ

l|

λ

p+1

.

Noticingthat lim n→+∞

1

−

ap+1

n

=

0

,

onecaneasilydeducefromEq.(27)that

Un_i

∼

n→+∞

α

(0) p+1

λ

n p+1vp+1

.

(28)

So,

Un_i behavesas

α

_p(0₊)₁

λ

n_p₊₁vp+1 whenthenumberofiterationsofthetimeintegrationn islarge.Moreover,thedifference

betweenthetruebehaviouranditsasymptoticapproximationdependsonhowtheratios

λ

l

/λ

p+1

forl

∈ {

1

,

...,

p

}

decay

to 0.Foralarge numberofiterations andforanyguess

U0 i with

α

(0)

p+1

=

0,

Uni behaves inthedirectionofthedominant

eigenvectorvp+1.

3.7.2. ResultsoftheMPMforSD

TheMatrixPowerMethodgivesawaytoperformthespectralanalysisoftheSpectralDifferencemethod.Thedispersion andtheampliﬁcationare simplygivenbythespectral radius

|λ

p+1

|

ofthematrixG.Betweentheiterations n andn

+

1,

thedispersionisgivenbytheargument

ϕ

= −

arg

λ

p+1

/

ν

andtheampliﬁcationisgivenby

ρ

=

λ

p+1

.

Remark. In all the cases presented in this paper, we found numericallythat the eigenvalue problem Gvl

= λ

lvl always

satisﬁedtheassumptionsgivenabove.Wedidnotsucceedindemonstratingmathematicallysucharesultforanyvalue of thepolynomialdegreep.Thisisleftasafuturelineofresearch.

Remark.ItshouldbenotedthattheMatrixPowerMethodcanbeappliedtoanykindofhigh-orderspectraldiscontinuous method,sincethediscretespace–timeintegrationcanbewrittenintheformofEq.(26)foranadvectionequation.

3.8. Comparisonwithnumericalresults

Numericalsolutionsin1DarenowcomputedtosupportthetheoreticalresultsondispersionanddissipationinFigs. 3a and 3b.TheadvectionequationEq.(1)withc

=

1

.

0 [m/s] issolvedusingtheSpectralDifferencemethodpreviously men-tionedwith

ν

=

0

.

1.Theinitialsolutionisu0

(

x

)

=

sin

(

kx

)

.Theone-dimensionalmeshiscomposedof40 regularcellswith

a cell length

x

=

2

.

0. Thesystemisclosed withperiodic boundaryconditions.Indeed,numericalcomputationsare per-formed withk

x

=

0

.

1

πn with

n

∈ N

,1

n

9 forp

=

2 and2

n

9 for p

=

3 andp

=

4.Oneobtains 2n periodson thisspeciﬁcmesh.

(12)

Fig. 4. SpectralanalysisfortheRKo6s-SDschemeswithCFLconditionν=0.1:comparisonoftheoreticalspectralbehaviourswithnumericalsolutionsand effectofdifferentpolynomialorders.

Fig. 5. Initial solution for kx=π/2.

Theinitialsolutionistransportedoverasufficientnumberofdiscretetime instantstomeasurethedissipationandthe dispersion.The amplificationandthephase shiftare identifiedbya minimizationprocess usingthe leastsquaresmethod tosolve:

min A∈R+∗

φ∈[0,2π[

||

f

(

x

,

t

)

−

A sin

(

kx

+ φ)||

₂

,

(29) where

·

2 is the standard L2 norm for functions. It is shown in Fig. 4a and Fig. 4b that theoretical and numerical

behavioursfordispersionanddissipationareinaverygoodagreement,forp

=

2, p

=

3 andp

=

4 respectively.

4. Extensiontohighwavenumbersforthespectraldifferencemethod

It was shownin Sec. 3.6that the standard spectral analysis, withFourier modes, cannot be applied to wavenumbers greaterthan

π

.ThisisaconsequenceoftheNyquist–Shannonsamplingtheorem.However,thespectralanalysispublished intheliteratureallowstocapturewavenumbersgreaterthan

π

[25–29,20,30].Thissectionisdevotedtotheanalysisofthis phenomenonwhenthewavenumberislargerthan

π

.

4.1. Aliasingandinitialsolutionprojection

Inthissection,wewanttoanalysethealiasingthatoccurswithhigh-orderspectralmethods.Weconsiderasbeforean advectionequationproblemwithperiodicconditions.Foragiven

x,twosimulationsareperformed.Fortheﬁrstone,the wavenumberk ischosen suchthatk

x

=

π

/

2,whileforthesecondcase,thewavenumberk issuchthatk

x

=

3

π

/

2.For bothcases,weselectp

=

3.

Fortheﬁrstcomputation,theinitial,ﬁnal solutions,andtheircorrespondingFouriertransform, areshowninFig. 5and

Fig. 6.Thesolutionisalmostconservedandtheenergyrepartitionpermodedoesnotchangesigniﬁcantly.

Regardingthesecondcasewithk

x

=

3

π

/

2,theinitialsolutionandtheassociatedFourierspectrumareshowninFig. 7. Aftermanyiterations(wedonotdeﬁnethenumberofiterationsheresinceitisthetopicofSec.4.3andSec.4.4),theﬁnal

(13)

Fig. 6. Final solution for kx=π/2.

Fig. 7. Initial solution for kx=3π/2.

Fig. 8. Final solution for kx=3π/2.

solutionandtheassociatedspectrumareshowninFig. 8.Weremarkthatthemodek

x

=

π

/

2 nowcontains thelargest partoftheenergy,whilethemodek

x

=

3

π

/

2 initiallyhadthelargestenergy.Allmodeslargerthan

π

aredampedandit remainsessentiallythemodeassociatedwithk

x

=

π

/

2.

This numericalexperimentpoints out the aliasingphenomenon revealedby our analysis. Suchaliasing occursforany valueof p,butthetimetoseeaneffectvarieswithp,thenumberofiterations,theCFLnumberandthewavenumberone looks for.Moreover, thestandardFourierspectral analysiscanonlybe appliedoncetheinitialwavenumberisunchanged: dissipation anddispersion aredeﬁnedforagivenfrequency.Here,sincethefrequency(withthemainpartoftheenergy) changes, it is mandatory to introduce a new wayto estimate dissipation anddispersion. In the following, dissipation is expressedusingrelationsdeﬁnedwithenergypreservingrelations,whiledispersionisobtainedfromascalarproduct.

(14)

4.2. Mathematicalconsideration

Letusdeﬁne the space L2

(

[

−

1

,

+

1]

)

ofcomplex-valued functionson the closedinterval

[−

1

,

+

1

]

andtheassociated complexscalarproductby:

∀ (

f

,

g

)

∈

L2

(

[

−

1

,

+

1]

)

2

,

f

,

g

=

+1

−1 f

(

x

)

×

conj

(

g

) (

x

)

dx

,

(30) whereconj

(

g

)

isthecomplexconjugateofg.Obviously,thenormassociatedwiththisscalarproductisdeﬁnedby:

∀

f

∈

L2

(

[

−

1

,

+

1]

) ,

f

=

+1

−1

|

f

(

x

)

|

2dx

.

(31)

Moreover,

α

ejβ_f

=

_α

_f

_and

_α

_ejβ_f

_,

_g

₌

_α

_ejβ

_f

_,

_g

_for_all

₍

_α

_{, β)}

_{∈ R}

+

_×

_[0

_,

₂

_π

_{[ by} _{sesquilinearity} _of _the_complex

scalarproduct.

LetususethiscomplexscalarproducttocomputethedispersionandthedissipationoftheSpectralDifferencemethod. Forwavenumbersin

[

0

,

π

]

,thespectral analysisfollowstheMPM.However,forwavenumberslargerthan

π

,comingback to the physical meaning ofdissipation anddispersion, dissipationcanbeexpressedastherateoflossinenergyofthesignal, whiledispersionisthephaseshiftofthesignal.Inthecaseofspectralanalysis,thefunctions f are simplyrepresentedbythe complexexponentialbasis fn

(

x

)

=

exp

(

−

jωn

t

+

jkx

)

=

α

ejβ_f0

₍

_x

₎

_._The_dissipation_and_the_dispersion _after_{n iterations}

canbeexpressedastheproductoftheinitialsolutionby

α

ejβ.Mathematically,thedissipationisdeﬁnedasthelossofthe

L2_-norm_of_the_signal _{f between}_time _{0 and}_{n by:}

fn

f0

=

α

ejβ_f0

f0

=

α

.

(32)

Thephaseshiftofthesignal f betweentime0 andn canbesimplyexpressedas: arg

fn

,

f0

=

arg

α

ejβf0

,

f0

=

arg

α

ejβ

f0

,

f0

=

arg

α

ejβ

f0

2

= β.

(33)

Thecomplexscalar productisdeﬁnedon L2

₍

_[

₋

₁

_,

₊

_1]

₎

_and_this_is_the _right_way_to_take _into_account _the_fact_that _all

quantities are definedby polynomials. However, Eq.(32) andEq.(33) are obtainedusing onlythe sesquilinearity ofthe complexscalarproductsincethedefinitionofthescalarproduct(Eq.(30))isneverexplicitlyused.Actually,thedefinition ofbothdispersionanddissipationdoesnotdepend ontheconsideredscalarproduct.Itismoreconvenient(andsimple!) toreplacetheL2 normforfunctionsbythe

2-norm

·

2 ofvectorsin

C

p+1,playingdirectlywiththesolutionsatsolution

points.Ofcourse,thisnormisderivedfromthefollowingcomplexscalarproduct:

∀ (

x

,

y

)

∈

C

p+1

2

_,

_x

_,

_y

₌

p+1

l=1 xl

×

conj

(

yl

).

(34)

4.3. Energylossestimation

Asexplainedbefore,wereplacethestandardL2-normoffunctionsbythe

2-normofthesolutionpointvectors

Un_i.Let usintroducetheratio

ρn

,mfor

(

n

,

m

)

∈ N

+

× N

+withn

>

m:

ρ

n,m

=

Un_i

₂

Un_i−m

₂

.

(35)

Bydeﬁnition,

ρn

,m representsthe energylossofthe solutionbetweeniterationn

−

m (n

≥

m) anditerationn. TheMPM

givesthebehaviourof

ρn

,m whenn issuﬃcientlylarge:

ρ

n,m

=

p+1 l=1

α

_l(0)

λ

n_l

2 p+1 l=1

α

_l(0)

λ

n_l−m

2

∼

n→+∞

λ

m p+1

∼

_n_→+∞

ρ

m

.

(36)

UsingEq.(36),itisclearthat

ρn

,m isageneralizationofthestandardcriterionforschemedissipationand

ρn

,m behavesas

(15)

Fig. 9. Loss in energy for two values of p as a function of the number of iterations (ν=0.1).

Let

ρn

=

ρ

n,nrepresentanestimationoftheenergylossbytheinitialsignalaftern iterations.InFig. 9,thelossinenergy

fortwovaluesofp isshownasafunctionofthenumberofiterationsforcomputationsperformedatCFLnumber

ν

=

0

.

1. Asexpected,thelossinenergyincreaseswiththenumberofiterationstoperform.

ρn

measuresthedissipationeffectabove

π

butweneedtointroduceanewquantitytoaccountfordispersion.

4.4. Phaseshiftestimation

Letusintroduce

δ

ϕ

n,mfor

(

n

,

m

)

∈ N

+

× N

+withn

>

m:

δ

ϕ

n,m

=

arg

Un_i

,

Un_i−mexp

(

−

jm

ν

k

x

)

.

(37)

δ

ϕ

n,m representsthephase shiftofthe solutionbetweeniterations n

−

m andn.Indeed,thefactorexp

(−

jmνk

x

)

takes

intoaccountthetheoreticaladvectiongivenbytheanalyticalsolutionoftheadvectionequationwithaconstantadvective velocity c.Wehave:

δ

ϕ

n,m

∼

n→+∞arg

α

_p(0₊)₁

λ

n_p₊₁conj

(

α

_p(0₊)₁

λ

n_p−₊m₁

)

exp

(

jm

ν

k

x

)

(38)

∼

n→+∞arg

α

(_p0₊₁)

2

ρ

nexp

(

−

jn

νϕ

)

ρ

n−mexp

(

j

(

n

−

m

)

νϕ

)

exp

(

jm

ν

k

x

)

(39)

∼

n→+∞arg

ρ

2n−m

α

(_p0₊)₁

2exp

(

jm

ν

(

k

x

−

ϕ

))

(40)

∼

n→+∞m

ν

(

k

x

−

ϕ

) .

(41)

This criterioncomputes thephaseshift ofthesignal inducedby thenumerical schemebetweeniterationn and iteration

n

−

m.Letusnote

δ

ϕ

n,n

= δ

ϕ

nwhichcomputesanestimationofthephasedelaybetweenthesignalaftern iterationsandthe

initialsignalwhichhasbeenanalyticallyconvected.Itmeasurestheoveralldispersioninducedbythetimeintegrationloop. Thislatterisageneralizationoftheusual criterionbutittakesintoaccount theeffectofthetime integration.Indeed,the spectral behaviourevolvesduringthenumberofiterations. Furthermore,

δ

ϕ

n

/

2

π

givesthephaseshiftlengthpernumber

ofwavelength.Forexample,

δ

ϕ

n

/

2

π

=

0

.

1 meansthatbothsignalsareseparatedby0

.

1

λ

where

λ

isthesignalwavelength.

In Fig. 10,thephaseshifts forSD3 andSD5are shownasa functionofthe numberofiterations,ina similarwayasfor

Fig. 9.

4.5. Summary

In this section, new criteriafor dissipation anddispersion havebeen introduced. The criteria take into account both space andtime schemes. It isshown that forwavenumbersabove

π

,aliasing occurs butthe aliasingspeed depends on theCFL number,thetime integrationprocedure,thenumberoftimesteps andthepolynomialdegree p. Thenewcriteria basedonenergylossandphaseshiftareextensionsofthestandardFourierapproachintroducedintheprevioussections.In particular,theresultspresentedinSec.3arerecoveredbythenewapproachforwavenumbersin

[

0

,

π

]

andalargenumber ofiterations(asymptoticbehaviour).Wenotethatthenewcriteriaaregeneralinthesensethattheycanbeappliedtoany kindofnumericalscheme.

In the next section, thesecriteria are usedto compare the Spectral Differencescheme withthe Finite Difference ap-proachesintroducedinSec.2.

(16)

Fig. 10. Phase shift for two values of p as a function of the number of iterations (ν=0.1).

Fig. 11. Spectral analysis forν=0.1: comparison of the schemes. For SD schemes, these results are obviously those given by the MPM. 5. Comparisonwithstandardhigh-orderschemes

Thissectionisdevotedtothecomparisonofthespectralbehaviourbetweenthestandardschemesforaeroacouticsand theSpectralDifferencemethod.

5.1. Naivecomparison

ForstandardspectralanalysisofFDschemes,

G (

k

x

)

deﬁnedinEq.(8)canbeseenasthetransferfunctionbetween

un i

and

un_i+1 afterapplyingthespatialnumericalscheme,thetimeintegrationandtheﬁlter.Likealltransferfunctions,itcan becharacterised byitsmodulus,whichwascalled

ρ

= |G|

,andits argument,whichwas called

ϕ

= −

arg

(G) /

ν

.Notethat the argumentof

G (

k

x

)

was (i) multipliedby

(−

1

)

because ofthenormalmode choice un

i

=

exp

(

(−

1

)

jn

ω

t

+

ji k

x

)

and (ii) divided by the CFL number

ν

to obtain a quantity which is always equal to k

x in the exact case since the dimensionlessdispersionrelationis

ω

t

=

νk

x.ForthespectralanalysisoftheSpectral Differencemethod,thetransfer functionis

λ

p+1.Indeed,whenthenumberofiterationsislarge,

U_in+1 isobtainedfrom

Un_i multipliedby

λ

p+1.Analogously,

ϕ

= −

arg

λ

p+1

/

ν

and

ρ

=

λ

p+1

weredeﬁned.ThesequantitiesareplottedinFig. 11aandFig. 11brespectively.

Inthelightoftheseﬁgures,theRKo6s–SD2schemeseemstohaveasimilarspectralbehaviour asbothRKo6s–CF8–CS6 andRKo6s–SFo11p–FDo11p.Neverthelessonehastorememberthatseveraldegreesoffreedomarelocatedwithinanymesh elementintheSpectralDifferenceapproach.Itisthereforepreferabletointroduceanewcriterionthat takesintoaccount thetotalnumberofdegreesoffreedominsteadofthenumberofmeshelementsandtheirsize.

5.2. Rescalingbythenumberofdegreesoffreedom

Forafaircomparison, theanalysismaybe performedconsideringthe dimensionlesswavenumberk

x builtwiththe mean distancebetween two degreesof freedom andnot with the element size

x.It meansthat, for the Spectral Dif-ferencemethod,thedimensionlesswavenumbershouldbek

x

/

(

p

+

1

)

whereas,forstandardschemes,thedimensionless wavenumberremainsk

x.Ofcourse,thesamerescalingisperformedfortheCFLnumber:

ν

= (

p

+

1

)

c

t

/

x for theSD method while

ν

=

c

t

/

x for FDschemes. Dispersion anddissipation are now deﬁned asa function of the numberof

(17)

Fig. 12. Spectral analysis forν=0.7: comparison of the considered schemes.

points per wavelength(PPW)andtwo deﬁnitionsareintroduced:PPW

= (

p

+

1

) λ/

x forthe SpectralDifferencemethod whilePPW

= λ/

x fortheFDschemes.Moreover,theCFLnumberiskeptconstant:

ν

=

0

.

7 forSDmethodand

ν

=

0

.

7 for standardschemes.ThischoiceismotivatedbythefactthatboththeSDandtheﬁnitedifferencemethodsleadtothesame physical time step.Moreover,the CFL valueis chosen inagreement withtheone forprevious aeroacousticcomputations usingthecompactscheme.

Finally,itmustbementionedthatthenewquantity

ν

|

k

x

−

ϕ

|

isintroducedtoaccountforthechangeinCFLnumber deﬁnitionforanalysingdispersion.

In termsofdispersion,theRKo6s–SD4 methodseems tobe equivalentto theRKo6s–CF8–CS6 combinationwhereas in termsofdissipation,theRKo6s–CF8–CS6combinationisbetweentheRKo6s–SD4andRKo6s–SD5method(Fig. 12).However, twoproblemscanbeobserved.First,thedimensionlesswavenumberbelongsto[0

,

π

/(

p

+

1

)

] fortheSDmethod.So,itcan not becomparedonthefullrangeofwavenumbersin[0

,

π

].Moreover,dispersion anddissipationin[0

,

π

/(

p

+

1

)

] come from the asymptotic behaviour introduced in Sec. 3.7.2 anddo not beneﬁt fromthe matrix form ofthe SD method for wavenumbershigherthan

π

/(

p

+

1

)

.Itisthusmandatorytoextendtheanalysisinenergylossandphaseshifttostandard schemesforafaircomparison.

5.3. Newcriteriaextensiontostandardaeroacoustic schemes

FortheconsideredFiniteDifferenceschemeswithtimeintegrationandﬁltering,thesolutionisupdatedusingasimple multiplication:

un_i+1

=

G

un_i (42)

Introducingasbeforethelossinenergyandthephaseshift,itcomes:

ρ

n

=

un_i

₂

u0_i

₂

=

G

n

_u0 i

2

u0_i

₂

= |

G

|

n

₌

_ρ

n

_.

₍₄₃₎ and

δ

ϕ

n

=

arg

un_iconj

(

u_i0exp

(

−

jn

ν

k

x

))

,

(44)

=

arg

G

n

_u0 iconj

(

u0iexp

(

−

jn

ν

k

x

))

,

(45)

=

arg

G

n_exp

₍

_jn

_ν

_k

_x

₎

,

(46)

=

n

ν

k

x

+

n arg

(

G

) ,

(47)

=

n

ν

(

k

x

−

ϕ

) .

(48)

(18)

Fig. 13. Spectral analysis forν=0.7: analysis of the dissipation-based criterion.

5.4. Pointsperwavelength(PPW)–dissipation-basedcriterion

IntroducingthenewcriteriaforFiniteDifferenceschemes,itisnowpossibletodeﬁneacut-offwavenumberkε_c

x that

isassociatedwiththeconservationofa certainpercentage1

−

ε

ofenergyaftern iterations.Todoso,itismandatoryto solvethefollowingproblem:

For a given set

(

n

,

ν

,

p

)

∈

N

∗

× R

+∗

× N

∗

, look for

k

x

∈

0

,

kε_c

x

such that

ρ

n

(

k

x

,

n

,

ν

,

p

)

1

−

ε

Thiscut-off wavenumber can be obviouslyinterpreted asa numberofpoints per wavelength (PPW).The dissipation-basedcriteriaisﬁnally showninFig. 13forbothFiniteDifferenceschemes,andfortheSpectralDifferenceapproachwith

p ranging from2 to 5 and n from104 to 107 atCFL number

ν

=

0

.

7.The cases withn

=

104 andn

=

105 correspond tothetypical numbersofiterations forstandardLES.Theextremecaseswithn

>

105 _are_only_computed_to_illustrate_the