hyperbolic conservation laws A new fifth order finite difference WENO scheme for solving Journal of Computational Physics

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Short note

A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws ^✩

Jun Zhu

^a

, Jianxian Qiu

^b,∗

aCollegeofScience,NanjingUniversityofAeronauticsandAstronautics,Nanjing,Jiangsu210016,PRChina

bSchoolofMathematicalSciencesandFujianProvincialKeyLaboratoryofMathematicalModelingandHigh-PerformanceScientific Computing,XiamenUniversity,Xiamen,Fujian361005,PRChina

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received16April2016

Receivedinrevisedform2May2016 Accepted3May2016

Availableonline7May2016

Keywords:

FifthorderWENOscheme Hyperbolicconservationlaws Finitedifferenceframework

Inthispaperanewsimplefifthorderweightedessentiallynon-oscillatory(WENO)scheme ispresentedinthefinitedifferenceframeworkforsolvingthehyperbolicconservationlaws.

ThenewWENOschemeisaconvexcombinationofafourthdegreepolynomialwithtwo linearpolynomialsinatraditionalWENOfashion.ThisnewfifthorderWENOschemeuses thesamefive-pointinformationastheclassicalfifthorderWENOscheme[14,20],couldget lessabsolutetruncationerrorsinL¹ andL^∞ norms,andobtainthesameaccuracyorder in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associatedlinearweightsareartificiallysettobeanyrandompositivenumberswiththe onlyrequirement that their sum equals one. New nonlinear weights are proposed for thepurpose ofsustainingthe optimalfifth orderaccuracy.The new WENOschemehas advantagesovertheclassicalWENOscheme[14,20] inits simplicityand easyextension tohigher dimensions. Some benchmark numerical tests are performed to illustrate the capabilityofthisnewfifthorderWENOscheme.

©^{2016ElsevierInc.Allrightsreserved.}

1. Introduction

Recently, manyhighorderfinitedifferenceorfinitevolume numericalmethods havebeeninvestigatedtosolveforhy- perbolic conservation laws. The essentially non-oscillatory(ENO) and weighted ENO (WENO) schemes, which have been appliedquitesuccessfullytosolvetheproblemswithstrongshocks,contactdiscontinuitiesandsophisticatedsmoothstruc- tures,aretheprimary schemesusedasthecurrentstate oftheartintheliterature. Forthepurposeofachievinguniform highorderaccuracybothinspaceandtime,HartenandOsher[11]gaveaweakerversionofthetotalvariationdiminishing (TVD)criterion [8]andon whichthey established abasis forthe reconstructionofhighorder ENOtype schemes. Anda series ofENO schemes were developed by Hartenet al.[10] to solve one dimensional problems.The mostcrucial spirit oftheseENOtype schemesistoapply thesmoothestcandidatestencilamongallstencilsto approximatethe variablesat cellboundariesforthesakeofsustaininghighorderaccuracyinsmoothregionandnotintroducingspuriousoscillationsin nonsmoothregion[21,22].ThefirstWENOschemewasoriginally proposedbyLiu,OsherandChan[15],inwhichinstead ofusingthe optimalsmooth candidatestencil,alinearconvexcombinationofallstencilsincludingnonsmoothstencils is used.Afterthat,suchWENOschemewasimprovedbyJiangandShu[14],inwhichageneralframeworkforthedesigning

✩ TheresearchispartlysupportedbyNSFCgrants11372005,11571290and91530107.

*

Correspondingauthor.

E-mailaddresses:[email protected](J. Zhu),[email protected](J. Qiu).

http://dx.doi.org/10.1016/j.jcp.2016.05.010 0021-9991/©^{2016ElsevierInc.Allrightsreserved.}

ofnewsmoothnessindicatorsandnonlinearweights wasspecified indetail.Forthesystemcase, theWENOschemesare basedonlocalcharacteristicdecompositionandnumericalfluxsplittingmethodtoavoidspuriousoscillationsnearbystrong shocksorcontactdiscontinuities. Andsome other famousWENOschemes areproposed forspecificalcomputation needs, suchasoptimizedWENOschemesforsolvinglinearwaveswithdiscontinuity[23],monotonicitypreservingWENOschemes forveryhighorderaccuracysimulation[2],hybridcompactWENOschemesforcomputingshockandturbulenceinteraction [17,18],androbustWENOschemesfordealingwithdistortedlocalmeshgeometryorthedegenerationofthemeshquality variesforcomplexcomputationaldomain[12,16,19]etc. ThevarioustypesofENOandWENOschemes[4,5,9,13–15,21,22, 25] are quitesuccessful innumericalsimulations forsteady andunsteady problemsharboring strongdiscontinuities and sophisticatedsmoothstructures.

Comparingwith the above mentioned literature, especially including classical WENO schemes proposed by Jiang and Shu[14,20], themajor advantages of thenewfinite difference fifth orderWENOscheme inthispaper are its simplicity, highorderaccuracyandeasyimplementationinthecomputation.ThisnewWENOschemehasaconvexcombinationofa fourthdegreepolynomialwhich shouldsubtractother twolinearpolynomialsbymultiplyingproper constantparameters, thencompoundthesethreeunequaldegreepolynomialswhichuseinformationondifferentspatialstencilsinaWENOtype methodologyincludingartificiallysettinglinearweightsfortheconservationwithoutlosingnumericalaccuracy,computing smoothnessindicatorsandproposingnewnonlinearweightformulawhichisdifferenttotheexpressionspecifiedin[3,6], forsolvingtheEulerequationsinone andtwodimensions. Theessentialmeritsofsuchmethodologyareits simplicityin spaceby thedefinition ofpositive linearweights, andonlyone five-pointstencil andtwotwo-point stencils are usedto reconstructthreedifferentdegreepolynomials. Aninnovationofmodifying thefourthdegreepolynomialis veryessential forsustainingthescheme’shighorderaccuracy,otherwise,sometraditionalrobustWENOmethodologieswilldegradetheir numericalaccuracytoalowerorderofaccuracy.Therefore,wetrytousefifthorderapproximationtothenumericalfluxfor simplicityandeasyimplementationforthesakeofobtaininghighorderaccuracyinsmoothregion,andswitchittoeither oftwosecondorderapproximationsonsmallerspatial stencilswhenthecomputingfield isadjacenttheshockorcontact discontinuities forthesake ofavoidingspuriousoscillations.Thereafter, thenewnonlinearweightformulasarepresented forthefinal highordernumerical fluxapproximation procedure.When alarge, centeredstencil permitsoptimalstability andaccuracytobereached,thismethodwillcertainlyreachthatstabilityandaccuracy.However,whenthesolutiononthat stencilisrough,itisbeneficialtoseekoutasmallerTVD-likestencil.Thetwosmallstencilsthatareusedensurethat(when thelargestencilmaybeignored)themethodiseffectivelyaTVDschemewithavanAlbada-likelimiter.Consequently,for non-smoothsolutionvector, thisschemeoperates likea TVDschemewithallthe attendant stabilityproperties ofaTVD scheme. In other words, the fact that a TVD-like property is built into the method makes it closer to a Monotonicity PreservingWENOthanapureWENO-JS(whichalwaysreliesonanensembleoflargestencils).Allinall,theinnovationsof thispaperlieinthreeaspects:thenewwayofreconstructingamodifiedfourthdegreepolynomialbysubtractingtwolinear polynomialswithproperparameters,anovelWENOtypecommunicationamongthreeunequal(modified)polynomialsfor highaccurateapproximationsinsmoothregionandnon-oscillationinnonsmoothregion,andthenewmannerofobtaining associatednonlinearweights.

Theorganizationofthepaperisasfollows:inSection2,weconstructthenewfinitedifferencefifthorderWENOscheme indetail.InSection 3,someclassical numericaltestsarepresentedtoverifythenumericalaccuracy andefficiencyofthe newWENOscheme.ConcludingremarksaregiveninSection4.

2. NewsimpleWENOscheme

Onedimensionalhyperbolicconservationlawsis

u_t

+

^fx

(

u

) =

⁰

,

u

(

x

,

0

) =

u0

(

x

),

^(2.1)

andtheassociatedsemidiscretizationof(2.1)canbereformulatedas du

dt

=

L

(

u

),

(2.2)

where L(u) isthe highorderspatial discrete formulationof −^fx(u). Forsimplicity,the uniformmeshis distributedinto some cells Ii= [^xi−¹/2,xi+¹/2]^{,withtheuniformcellsizeisdenotedasx}i+¹/2−^xi−¹/2=^{handassociatedcellcentersare} definedasxi=¹₂(xi+¹/2+^xi−¹/2).ui(t)isdefinedasanodalpointvalueu(xi,t).Thereforetherighthandsideof(2.2)can bereformulatedas

L

(

ui

(

t

)) = −

¹

h

( ˆ

_fi+1/2

− ˆ

fi−¹/2

),

(2.3)

where ˆfi+¹/2 isanumericalfluxwhichhasafifthorderapproximationofflux f(u)attheboundaryxi+¹/2oftargetcellIi inthispaper.Ifthenumericalflux ˆfi+¹/2istakenintoaccountthefifthorderapproximation,then ¹_h(ˆfi+¹/2− ˆ^fi−¹/2)isthe fifthorderapproximationto fx(u)atx=^xi.Foranordinaryflux f(u),itcanbesplitinto f(u)= ^f+(u)+^f−(u),satisfying

df⁺(u)

du ≥^{0 and df}_du^−(u)≤^{0.A simplest} Lax–Friedrichs splittingis applied as f^±(u)= ¹₂(f(u)±

α

u), inwhich

α

is set as

maxu|^f(u)|^{overthewholerangeofu.ThedetailedflowchartofthenewsimplefifthorderWENOschemeisdescribedas} follows,forsimplicity,weonlydothereconstructionof f⁺(u)atpointxi+¹/2anddenoteitas ˆf_i⁺_+1/2.

Step1.Choosethefollowingbigstencil:T₁= {^Ii−²,I_i−1,I_i,I_i+1,I_i+2}^{.Itiseasytoobtainafourthdegree}reconstructed polynomial p1(x)whichisbasedonthenodalpointinformationofthefluxsplittingandsatisfying:

1 h

x_j

+^h/2 xj−h/2

p₁

(

x

)

dx

=

^f+_j

,

j

=

ⁱ

−

²

,

i

−

¹

,

i

,

i

+

¹

,

i

+

²

.

(2.4)

Itsexplicitexpressionissameasspecifiedin[1]

p₁

(

x

) =

^f_i⁺

+ −

82f_i⁺₋₁

+

11f_i⁺₋₂

+

82f_i⁺₊₁

−

11f_i⁺₊₂

120

(

^x

−

^xi

h

) +

40f_i⁺₋₁

−

^3f_i⁺₋₂

−

^74f_i⁺

+

^40f_i⁺₊₁

−

^3f_i⁺₊₂

56

((

^x

−

^xi

h

)

²

−

¹ 12

) +

2f_i⁺₋₁

−

^f_i⁺₋₂

−

^2f_i⁺₊₁

+

^f_i⁺₊₂

12

((

^x

−

xi

h

)

³

−

³ 20

(

^x

−

xi

h

)) +

−

4f_i⁺₋₁

+

f_i⁺₋₂

+

6f_i⁺

−

4f_i⁺₊₁

+

f_i⁺₊₂

24

((

^x

−

^xi

h

)

⁴

−

³ 14

(

^x

−

^xi

h

)

²

+

³

560

).

(2.5)

Choose another two smaller stencils: T2= {^Ii−¹,Ii} ^{and T}3= {^Ii,Ii+¹}^{.It isalso very easy forus to get the two linear} polynomialswhicharebasedonthenodalpointinformationofthefluxsplittingandsatisfying

1 h

xj

+h/2 x_j−^h/2

p₂

(

x

)

dx

=

^f+_j

,

j

=

ⁱ

−

¹

,

i

,

(2.6)

and 1 h

xj

+h/2 x_j−^h/2

p3

(

x

)

dx

=

^f+_j

,

j

=

ⁱ

,

i

+

¹

.

(2.7)

Theirexplicitexpressionsare

p₂

(

x

) =

^f_i⁺

+ (

f_i⁺

−

^f_i⁺₋₁

)(

^x

−

^xi

h

),

(2.8)

and

p3

(

x

) =

f_i⁺

+ (

f_i⁺₊₁

−

f_i⁺

)(

^x

−

^xi

h

).

(2.9)

Step 2. The main selection principle ofthe linear weights is solelybased on the consideration ofa balance between accuracy and ability to achieve essentially nonoscillatory shock transitions. In all of our numerical tests, following the practice in[7,26],wetake thepositivelinearweights as

γ

1=⁰.98 and

γ

2=

γ

3=⁰.01.Thelinearweightscan bechosen to beanysetofpositivenumberswiththecondition thatthesummationisoneandwouldnotpollutethenewscheme’s optimalaccuracy.

Step3.Computethesmoothnessindicatorsβn, n=¹,2,3,whichmeasurehowsmooththefunctionspn(x), n=¹,2,3, are inthetarget cell Ii.The smallerthesesmoothnessindicators,the smootherthefunctionsarein Ii.We usethesame recipeforthesmoothnessindicatorsasin[1,14,20]

β

n

=

r α=¹

Ii

h^2α−1

(

^d

αpn

(

x

)

dx^α

)

²dx

,

n

=

1

,

2

,

3

.

(2.10)

Theassociatedexplicitexpressionsare

β

1

= ( −

^82f_i⁺₋₁

+

^11f_i⁺₋₂

+

^82f_i⁺₊₁

−

^11f_i⁺₊₂

120

+

2f_i⁺₋₁

−

^f_i⁺₋₂

−

^2f_i⁺₊₁

+

^f_i⁺₊₂

120

)

²

+

13 3

(

^40f

i+−1

−

3f_i⁺₋₂

−

74f_i⁺

+

40f_i⁺₊₁

−

3f_i⁺₊₂

56

+

123 455

−

^4f_i⁺₋₁

+

^f_i⁺₋₂

+

^6f_i⁺

−

^4f_i⁺₊₁

+

^f_i⁺₊₂

24

)

²

+

781 20

(

^2f

+

i−1

−

^f_i⁺₋₂

−

^2f_i⁺₊₁

+

^f_i⁺₊₂

12

)

²

+

1421461

2275

( −

^4f_i⁺₋₁

+

^f_i⁺₋₂

+

^6f_i⁺

−

^4f_i⁺₊₁

+

^f_i⁺₊₂

24

)

²

,

(2.11)

β

2

= (

f_i⁺₋₁

−

f_i⁺

)

²

,

(2.12)

and

β

₃

= (

f_i⁺

−

^f_i⁺₊₁

)

²

.

(2.13)

Theexpansionsof(2.11)to(2.13)inTaylorseriesabout f_i⁺areobtainedas

β

1

=

^h2

((

f_i⁺

)

)

²

+

^13h4

((

f_i⁺

)

)

²

/

12

+

^h6

(

5467

((

f_i⁺

)

)

²

−

¹⁴

(

f_i⁺

)

(

f_i⁺

)

⁽⁴⁾

−

³³⁶

(

f_i⁺

)

(

f_i⁺

)

⁽⁵⁾

)/

5040

+

^O

(

h⁸

) =

^h2

((

f_i⁺

)

)

²

(

1

+

^O

(

h⁴

)) =

^O

(

h²

),

(2.14)

β

2

=

h²

((

f_i⁺

)

)

²

−

h³

(

f_i⁺

)

(

f_i⁺

)

+

h⁴

( (

f_i⁺

)

4

+ (

f_i⁺

)

(

f_i⁺

)

3

) +

O

(

h⁵

) =

h²

((

f_i⁺

)

)

²

(

1

+

^O

(

h

)) =

^O

(

h²

),

(2.15)

and

β

3

=

^h2

((

f_i⁺

)

)

²

+

^h3

(

f_i⁺

)

(

f_i⁺

)

+

^h4

( (

f_i⁺

)

4

+ (

f_i⁺

)

(

f_i⁺

)

3

) +

^O

(

h⁵

) =

h²

((

f_i⁺

)

)

²

(

1

+

^O

(

h

)) =

^O

(

h²

).

(2.16)

Itisassumedthat theindicatorsofsmoothnesscanbe rewrittenas:β1=^D(1+^O(h⁴))andβ2,3=^D(1+^O(h)),inwhich D=^h2((f_i⁺))² isanon-zeroconstantindependentofnonconditionthat(f_i⁺)=^0.

Step 4. Calculate the non-linear weights based on the linearweights and the smoothnessindicators. Forinstance, as shownin[3,6],weusenew

τ

whichissimplydefinedastheabsolutedeferencebetweenβ1,β2 andβ3,andisdifferentto theformulaspecifiedin[3,6].SincethetwodifferenceexpansionsinTaylorseriesabout f_i⁺are

β

1

− β

2

= (

f_i⁺

)

(

f_i⁺

)

h³

+ (

⁵

((

f_i⁺

)

)

²

6

− (

f_i⁺

)

(

f_i⁺

)

3

)

h⁴

+

^O

(

h⁵

)

=

O

(

h³

), (

f_i⁺

)

=

⁰

, (

f_i⁺

)

=

⁰

,

O

(

h⁴

), (

f_i⁺

)

=

⁰

, (

f_i⁺

)

=

⁰

,

^(2.17)

and

β

1

− β

3

= − (

f_i⁺

)

(

f_i⁺

)

h³

+ (

⁵

((

f_i⁺

)

)

²

6

− (

f_i⁺

)

(

f_i⁺

)

3

)

h⁴

+

^O

(

h⁵

)

=

O

(

h³

), (

f_i⁺

)

=

⁰

, (

f_i⁺

)

=

⁰

,

O

(

h⁴

), (

f_i⁺

)

=

0

, (

f_i⁺

)

=

0

.

^(2.18)

Itiseasytoverify

τ = ( |β

1

− β

2

| + |β

1

− β

3

|

2

)

²

=

O

(

h⁶

), (

f_i⁺

)

=

⁰

, (

f_i⁺

)

=

⁰

,

O

(

h⁸

), (

f_i⁺

)

=

⁰

, (

f_i⁺

)

=

⁰

.

^(2.19)

Thenwedefine

ω

n

= ω ¯

n

₃

=¹

ω ¯

^, ω ^¯

n

= γ

n

(

1

+ τ ε + β

n

),

n

=

1

,

2

,

3

.

(2.20)

Here

ε

isasmallpositivenumbertoavoidthedenominatortobecomezero.Bytheusageof(2.14)to(2.16) and(2.19) in thesmoothregion,itsatisfies

τ

ε + β

n

=

O

(

h⁴

),

n

=

1

,

2

,

3

,

(2.21)

oncondition that

ε

βn.Therefore,thenonlinearweights

ω

n, n=¹,2,3,satisfytheorderaccuracycondition

ω

n=

γ

n+ O(h⁴)[3,6],providingtheformalfifthorderaccuracytotheWENOschemeelaboratedin[14,20].Wetake

ε

=^{10−6 inour} computation.

Remark.Forcase(f_i⁺)=^0,(f_i⁺)=^{0,itisalsoeasytoverifythat}

ω

n=

γ

n+^O(h⁴). Step5.Thenewfinalreconstructionsofthenumericalfluxatx=^xi+¹/2isgivenby

ˆ

f_i⁺_+1/2

= ω

1

(

¹

γ

1

p₁

(

x_i+1 2

) − γ

2

γ

1

p₂

(

x_i+1 2

) − γ

3

γ

1

p₃

(

x_i+1

2

)) + ω

2p₂

(

x_i+1

2

) + ω

3p₃

(

x_i+1

2

).

(2.22)

The first termof the right hand side of (2.22) looks very complicatedat the first glance. If (2.22) is simply definedas

ω

1p1(x_i+1

2)+

ω

2p2(x_i+1

2)+

ω

3p3(x_i+1

2)as usual, the schemewould degrade its optimalfifth order accuracy because of p2(x_i+1

2)andp3(x_i+1

2)areactiveandtheconvexcombinationtogetherwithp1(x_i+1

2)wouldnotofferhighorderapproxi- mation atpoint x_i+1/2 tonumericalfluxinsmoothregion.Thusweshoulddosomethingtowipeofftheircontributionin smoothregion.Bydoingso,ifabigspatialstencilpermitsoptimalstabilityandaccuracytobereached,(2.22)couldobtain thatstability andaccuracyobviously.However,whenthesolutiononthebigspatialstencilisrough,itisbeneficialtoseek outsmallertwo-pointTVD-likestencils.Suchtwosmallspatialstencilsthatareusedensurethat(whenthelargefive-point spatial stencil may be inactive andignored) the equation (2.22) can gradually transit to become a TVD scheme with a vanAlbada-likelimiter.So,forthepurposeofsolvingnon-smoothnumericalsolutions,(2.22)performs likeaTVDscheme withalltheattendant stabilitypropertiesofaTVDscheme.Inthisway,itisnotverycrucialtodeliberatelychooselinear weights forthepurposeofsustaininghighorderaccuracyinsmoothregionandcouldkeeptheshocktransitionsharplyin nonsmoothregion.

Step 6.Thesemidiscretescheme(2.2)isdiscretizedintimebyathirdorderTVDRunge–Kuttamethod[21]

⎧ ⎪

⎨

⎪ ⎩

u⁽¹⁾

=

^un

+

^{t L}

(

uⁿ

),

u⁽²⁾

=

³₄^un

+

¹₄^u(1)

+

¹₄ ^{t L}

(

u⁽¹⁾

),

uⁿ⁺¹

=

¹₃^un

+

²₃^u(2)

+

²₃ ^{t L}

(

u⁽²⁾

).

(2.23)

Remark. Forthe systemsofconservationlaws,such asthecompressibleEulerequations,all ofthereconstruction proce- duresare implementedinthelocalcharacteristicdirectionsforthepurposeofavoidingspurious oscillations.Forthetwo dimensionalproblems,allofthesereconstructionproceduresarecarriedoutina dimension-by-dimensionfashion.

3. Numericaltests

In this section we presentthe results ofnumerical tests ofthe fifth ordernew WENO schemetermed as WENO-ZQ specified intheprevious section incomparisontotheclassical WENOschemetermedasWENO-JS narratedin[14,20] in one andtwodimensions. TheCFLnumberissetas0.6forWENO-JSandWENO-ZQschemesinourcomputations.Forthe purposeoftestingwhethertherandomchoiceofthelinearweightswouldpollutetheoptimalorderaccuracyofWENO-ZQ scheme ornot, we set differenttype of linearweights in thenumerical accuracycases as:(1)

γ

1=⁰.98,

γ

2=⁰.01 and

γ

3=⁰.01;(2)

γ

1=¹.0/3.0,

γ

2=¹.0/3.0 and

γ

3=¹.0/3.0;(3)

γ

1=⁰.01,

γ

2=⁰.495 and

γ

3=⁰.495.Andset

γ

1=⁰.98,

γ

2=⁰.01 and

γ

3=⁰.01 inthelatterexamples.

Example3.1.WesolvethefollowingnonlinearscalarBurgersequation:

μ

t

+ ( μ

²

2

)

x

=

⁰

,

0

<

x

<

2

,

(3.1)

with the initial condition

μ

(x,0)=⁰.5+^sin(

π

x)and periodic boundary condition. When t=⁰.5/

π

the solution isstill smooth,andtheerrorsandnumericalordersofaccuracybytheWENO-ZQschemeareshowninTable 3.1.Forcomparison, errors andnumericalordersofaccuracy bythe classicalWENO-JS schemeare showninthesame table.We canseethat bothWENO-ZQandWENO-JSschemesachievetheirdesignedorderofaccuracy,andWENO-ZQschemewithdifferenttype of linearweights produces lesstruncationerrors andachieves optimalorders ofaccuracy. Fig. 3.1 showsthat WENO-ZQ schemeneedslessCPUtimethanWENO-JS doestoobtainthesamequantitiesof L¹ andL^∞ errors,soWENO-ZQscheme ismoreefficientthanWENO-JSschemeinthistestcase.

Example3.2.Wesolvethefollowingtwo-dimensionalnonlinearscalarBurgersequation:

μ

t

+ ( μ

²

2

)

x

+ ( μ

²

2

)

y

=

⁰

,

0

<

x

,

y

<

4

,

(3.2)

withtheinitialcondition

μ

(x,y,0)=⁰.5+^sin(

π

(x+^y)/2)andperiodicboundaryconditions.Whent=⁰.5/

π

thesolution isstillsmooth,andtheerrorsandnumericalordersofaccuracybytheWENO-ZQschemeincomparisontothatofWENO-JS

Table 3.1

μt+(^μ₂²)x=^{0.Initialdata}μ(x,0)=⁰.5+^sin(πx).WENO-ZQschemeandWENO-JSscheme.T=⁰.5/π.L¹andL^∞errors.

Grid points WENO-ZQ (1) scheme WENO-JS scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10 1.64E−^{2 5.32E}−^{2 1.91E}−^{2 7.48E}−²

20 1.44E−3 3.51 9.73E−3 2.45 2.06E−3 3.21 1.21E−2 2.63

40 7.29E−5 4.31 6.93E−4 3.81 1.24E−4 4.05 1.03E−3 3.54

80 2.38E−6 4.93 3.06E−5 4.50 4.41E−6 4.81 4.72E−5 4.46

160 7.07E−^{8 5.07 9.31E}−^{7 5.04 1.64E}−^{7 4.75 1.38E}−^{6 5.09}

320 2.09E−^{9 5.07 2.78E}−^{8 5.06 4.76E}−^{9 5.11 7.28E}−^{8 4.25}

Grid points WENO-ZQ (2) scheme WENO-ZQ (3) scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10 3.48E−^{2 1.10E}−^{1 3.76E}−^{2 1.16E}−¹

20 4.55E−^{3 2.93 1.55E}−^{2 2.83 5.61E}−^{3 2.74 1.95E}−^{2 2.58}

40 1.41E−^{4 5.01 8.15E}−^{4 4.25 1.68E}−^{4 5.06 1.13E}−^{3 4.11}

80 2.57E−6 5.77 3.05E−5 4.74 2.68E−6 5.97 3.04E−5 5.21

160 7.10E−8 5.18 9.31E−7 5.03 7.12E−8 5.23 9.31E−7 5.03

320 2.09E−9 5.08 2.78E−8 5.06 2.09E−9 5.09 2.78E−8 5.06

Fig. 3.1.μt+(^μ₂²)x=0.Initialdataμ(x,0)=0.5+sin(πx).Computingtimeanderror.NumbersignsandasolidredlinedenotetheresultsofWENO-ZQ schemewithdifferentlinearweights(1),(2)and(3);squaresandasolidlinedenotetheresultsofWENO-JSscheme.(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

schemeareshowninTable 3.2.WecanseethattheWENO-ZQschemewithdifferenttypeoflinearweightsachievesclose to its designed order of accuracy and generates less absolute truncation errors. And the Fig. 3.2 shows that WENO-ZQ schemeneedslessCPUtimethanWENO-JSschemedoestoobtainthesamequantitiesofL¹ andL^∞errors.

Example3.3.

∂

∂

t

⎛

⎝ ρ ρμ

E

⎞

⎠ + ∂

∂

x

⎛

⎝ ρμ ρμ

²

+

p

μ (

E

+

^p

)

⎞

⎠ =

⁰

.

(3.3)

In which

ρ

is density,

μ

is the velocity in x-direction, E is total energy and p is pressure. The initial conditions are:

ρ

(x,0)=¹+⁰.2sin(x),

μ

(x,0)=^1,p(x,0)=^1,

γ

=¹.4.Thecomputingdomainisx∈ [⁰,2

π

]^{.Periodicboundarycondition} isappliedinthistest.Theexactsolutionis

ρ

(x,t)=¹+⁰.2sin(x−^t).Thefinaltimeist=^{2.Theerrorsandnumericalorders} ofaccuracyofthedensitybytheWENO-ZQschemeandWENO-JSschemeareshowninTable 3.3andthenumericalerror againstCPUtime graphsareinFig. 3.3.Wecanobservethat thetheoreticalorderis actuallyachievedandthe WENO-ZQ schemecangetbetterresultsandismoreefficientthanWENO-JSschemeinthistestcase.

Table 3.2

μt+(^μ₂²)x+(^μ₂²)y=^{0.Initialdata}μ(x,y,0)=⁰.5+^sin(π(x+^y)/2).WENO-ZQschemeandWENO-JSscheme.T=⁰.5/π.L¹andL^∞errors.

Grid points WENO-ZQ (1) scheme WENO-JS scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10×^{10 1.84E}−^{2 5.38E}−^{2 2.06E}−^{2 7.49E}−²

20×20 1.73E−3 3.41 9.49E−3 2.50 2.16E−3 3.25 1.21E−2 2.63

40×40 7.46E−5 4.54 6.93E−4 3.78 1.27E−4 4.09 1.03E−3 3.54

80×80 2.41E−6 4.95 3.06E−5 4.50 4.47E−6 4.83 4.72E−5 4.46

160×^{160 7.11E}−^{8 5.08 9.31E}−^{7 5.04 1.65E}−^{7 4.76 1.38E}−^{6 5.09}

320×^{320 2.09E}−^{9 5.08 2.78E}−^{8 5.06 4.77E}−^{9 5.11 7.28E}−^{8 4.25}

Grid points WENO-ZQ (2) scheme WENO-ZQ (3) scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10×^{10 3.81E}−^{2 1.10E}−^{1 4.11E}−^{2 1.17E}−¹

20×^{20 4.82E}−^{3 2.98 1.56E}−^{2 2.82 5.85E}−^{3 2.81 1.92E}−^{2 2.60}

40×^{40 1.42E}−^{4 5.08 7.94E}−^{4 4.30 1.70E}−^{4 5.10 1.10E}−^{3 4.12}

80×80 2.61E−6 5.77 3.05E−5 4.70 2.71E−6 5.97 3.04E−5 5.18

160×160 7.14E−8 5.19 9.30E−7 5.03 7.17E−8 5.24 9.30E−7 5.03

320×320 2.09E−9 5.09 2.78E−8 5.06 2.09E−9 5.09 2.78E−8 5.06

Fig. 3.2.μt+(^μ₂²)x+(^μ₂²)y=0.Initialdataμ(x,y,0)=0.5+sin(π(x+y)/2).Computingtimeanderror.Numbersignsandasolidredlinedenotethe resultsofWENO-ZQschemewithdifferentlinearweights(1),(2)and(3);squaresandasolidlinedenotetheresultsofWENO-JSscheme.(Forinterpretation ofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Table 3.3

1D-Eulerequations:initialdataρ(x,0)=¹+⁰.2sin(x),μ(x,0)=^{1 andp}(x,0)=^{1.WENO-ZQschemeandWENO-JSscheme.T}=^{2.L1andL∞errors.}

Grid points WENO-ZQ (1) scheme WENO-JS scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10 1.36E−3 3.51E−3 4.49E−3 6.79E−3

20 3.04E−^{5 5.49 5.97E}−^{5 5.87 2.15E}−^{4 4.38 3.75E}−^{4 4.18}

40 9.62E−^{7 4.98 1.60E}−^{6 5.21 6.74E}−^{6 5.00 1.28E}−^{5 4.87}

80 3.01E−^{8 5.00 4.75E}−^{8 5.07 2.07E}−^{7 5.02 3.99E}−^{7 5.01}

160 9.39E−^{10 5.00 1.47E}−^{9 5.01 6.38E}−^{9 5.02 1.15E}−^{8 5.11}

320 2.93E−11 5.00 4.60E−11 5.00 1.92E−10 5.05 3.16E−10 5.19

Grid points WENO-ZQ (2) scheme WENO-ZQ (3) scheme

L¹error Order L^∞error Order L¹error Order L^∞error Order

10 8.59E−3 2.27E−2 1.08E−2 2.70E−2

20 1.30E−4 6.05 3.98E−4 5.84 1.88E−4 5.85 5.31E−4 5.67

40 9.98E−^{7 7.02 5.07E}−^{6 6.29 1.15E}−^{6 7.34 6.80E}−^{6 6.28}

80 3.01E−^{8 5.05 6.48E}−^{8 6.29 3.01E}−^{8 5.27 7.35E}−^{8 6.53}

160 9.39E−^{10 5.00 1.54E}−^{9 5.39 9.39E}−^{10 5.00 1.58E}−^{9 5.53}

320 2.93E−^{11 5.00 4.60E}−^{11 5.07 2.93E}−^{11 5.00 4.61E}−^{11 5.10}