Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing

Applied Mathematics and Computation 403 (2021) 126131

ContentslistsavailableatScienceDirect

Applied Mathematics and Computation

journalhomepage:www.elsevier.com/locate/amc

Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing

Sergio Amat

^a,1

, Juan Ruiz-Álvarez

^a,1,∗

, Chi-Wang Shu

^b,2

, Dionisio F. Yáñez

^c,3

a Department of Applied Mathematics and Statistics, Universidad Politécnica de Cartagena, (Spain)

b Division of Applied Mathematics, Brown University, USA

c Department of Mathematics, Universidad de Valencia, Spain

a rt i c l e i nf o

Article history:

Received 1 August 2020 Revised 17 February 2021 Accepted 21 February 2021 Available online 21 March 2021 JEL classification:

65D05 65D17 65M06 65N06 Keywords:

WENO Cell-average New optimal weights Multiresolution schemes

Improved adaption to discontinuities Signal processing

a b s t ra c t

Inthispaperwetranslatetothecell-averagesettingthealgorithmforthepoint-valuedis- cretizationpresentedinAmatelal.(2020).Thisnewstrategytriestoimprovetheresults ofWENO-(2r− 1)algorithmclosetothesingularities,resultinginanoptimalorderofac- curacyatthesezones.Themainideaistomodifytheoptimalweightssothattheyhavea nonlinearexpressionthatdependsonthepositionofthediscontinuities.Inthispaperwe study theapplicationofthenewalgorithmtosignalprocessingusingHarten’smultires- olution.Severalnumericalexperimentsareperformedinordertoconfirmthetheoretical resultsobtained.

© 2021 The Authors. Published by Elsevier Inc.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introductionandreview:Harten’smultiresolutionforcell-averagesetting

Multiresolutionand, inparticularHarten’smultiresolution(MR),hasbeenusedextensivelyforacademicandindustrial applications inthepast years.Someexamples aresignal andimage processing, compressionordenoising(see,e.g. [2,3]).

Thereexistsaninterestingconnectionbetweenwaveletstheory,subdivisionschemesandMR(see[4]).Whiletheapproach presentedinwaveletstheory isbasedonfunctionalanalysis,MRallows toanalyzeandstudytheproblemfromthepoint

∗Corresponding author.

E-mail addresses: [email protected] (S. Amat), [email protected] (J. Ruiz-Álvarez), [email protected] (C.-W. Shu), [email protected] (D.F.

Yáñez).

1 This work was funded byproject 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia) and through the national research project (MINECO/FEDER) PID2019-108336GB-I00.

2 The author has been supported through the NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055.

3 The author has been supported through the Spanish MINECO project MTM2017-83942-P.

https://doi.org/10.1016/j.amc.2021.126131

0 096-30 03/© 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

of view offunction approximationandinterpolation. The field isalsoconnected tosubdivision schemes,that are usually appliedincomputeraideddesignandthatcanbemodifiedtocreateaMRschemeundersometheoreticalconditions.

Harten’sMRconsistsoffouroperators:letussupposethatl isthelevelofresolutionandthat

{

^xlj

}

²_j=0^{l areequallyspaced}

pointsof theinterval [0,1]withx^l_j=j/2^l,h_l=1/2^l.We interpretthe dataasthediscretization ofan integrablefunction, L¹(^[0,1]),throughthediscretizationoperator:

Dl:L¹

(

^[0,1]

)

^→Vl,

whereV^l isadiscretespace.Intheliterature,severaldiscretizationoperatorscanbefound(see,e.g.[2,3,5,6]).Inthispaper, weconsiderourdataastheaveragesofafunctionintheintervalsI^l_j=[(^j− 1)· hl,j· hl],i.e.eachcomponentof ¯f^lisdefined as:

¯f^l_j:=

(

Dlf

)

j=h⁻¹_l

_x^l

j x^l_j−1

f

(

^x

)

^dx, j=1,...,2^l. (1)

Oncewehavechosenthediscretization,inordertomovebetweentwoconsecutivelevels,twooperatorsaredefined:deci- mationandprediction.Thedecimationoperator,

D^l_l⁻¹:V^l →V^l−1,

allowstogofromafinerresolutiontoacoarserresolutionandisalinearoperator.Inourcase,wecandefineitusingthe linearityoftheintegral.Thus,for j=1,...,2^l−1,wehavethat,

(

D^l−1_l ¯f^l

)

j=1

2

(

^¯f2^lj−1+ ¯f₂^l_j

)

=2^l 2



_{ x}l 2j−1 x^l_2j−2

f

(

^x

)

^dx+

 x^l_2j x^l_2j−1

f

(

^x

)

^dx



=2^l−1

 x^l−1_j x^l−1_j−1

f

(

^x

)

^dx= ¯f^l−1_j . (2)

Now we haveto design apredictionoperator that allows toapproximatethe valuesatthe levell from dataatthe level l− 1.Wedenoteitas:

P_l^l₋₁:V^l−1→V^l.

Inordertoconstructthisoperator,weneedtoimposeaconsistencyproperty:

D^l_l⁻¹◦ P_l^l₋₁=IV^l−1, (3)

beingI_Vl−1 theidentity functioninthesubspaceV^l−1.Thisisa crucialpropertybecauseitallowsto recoverall theinfor- mationcontainedintheoriginaldataifwefirstlyapplythepredictionoperatorandthenthedecimationoperator.

Followingthepreviousdefinitions,P_l^l₋₁¯f^l−1isanapproximationof ¯f^l andwecandefinetheerrorvectorasthedifference betweentheoriginaldataandtheapproximation,

e^l_j= ¯f^l_j−

(

P_l^l₋₁¯f^l−1

)

j, j=1,...,2^l.

Then,usingEqs.(2)and(3),foranyj=1,...,2^l−1,wehavethat

(

D^l−1_l e^l

)

j =

(

D^l−1_l ¯f^l

)

j−

(

D_l^l−1P_l−1^l ¯f^l−1

)

j, 1

2

(

^el2j+e^l_2j−1

)

= ¯f^l−1_j − ¯f^l−1_j , e^l_2j+e^l_2j−1 =0.

(4)

Wecandefinethevectorofdetailsasthenonredundantinformationcontainedinthevectoroferrors.Wecandenotethe vectorofdetailsbyd^l_j=e^l_2j−1, j=1,...,2^l−1.Therefore,ifListhefinestlevelofresolution,wecanobtainaMRrepresen- tationof ¯f^L usingthepreviouslypresentedoperatorsattheLscalesofresolution:

¯f^L≡

(

^¯f0,d¹,...,d^L

)

.

The processingofthevectorsofdetails ateach scaleofresolution allowsto designseveralapplications.Forexample,the truncation of the details using hard thresholding provides compression, the truncation usingsoft thresholding allows to reducetextureornoise,etc.

Hence,ourprincipalobjectiveistoconstructapredictionoperatorwhichadequatelyapproximatestheoriginaldata.In the literature, severalpredictionoperators based onlinearandnonlinear interpolation techniquescan be found(see, e.g.

[2,3,7,8]).Atthispoint,itisnecessarytodesignareconstructionoperatorthatallowstoobtainanintegrablefunctionfrom thedataatthelevell,i.e.

Rl:V^l→L¹

(

^[0,1]

)

^.

Taking thisinto account, we can define the predictionoperator asthe composition of the decimation operator andthe reconstructionoperator,

P_l^l₋₁:=Dl◦ R_l−1. (5)

S. Amat, J. Ruiz-Álvarez, C.-W. Shu et al. Applied Mathematics and Computation 403 (2021) 126131

LinearpiecewiseinterpolationhasbeenusuallyappliedasthereconstructionoperatorRl,(see,e.g.[2,3,7,8]).However,this type ofinterpolation produces Gibbs phenomenon at discontinuityzones. Inorder to avoid thisphenomenon, nonlinear techniques havebeenintroducedinthe pastyears.Essentiallynon oscillatory(ENO)method (seee.g. [3,7,9–11]) haspro- duced interesting results when it has been introduced asthe reconstruction operator. In [12],the authors proposed the WENO(WeightedENO)algorithmwiththeaimofobtainingahigherorderofaccuracyatsmoothzoneswhilekeepingthe adaptionpropertiesofENOmethodclosetothediscontinuities.WENOmethodconsistsofanonlinearconvexcombination oftheinterpolantsconstructedusingthedifferentstencilsthatENOalgorithmconsiders.Forthis,asetofnonlinearweights are designedby meansofsome values calledsmoothnessindicators.Thesesmoothnessindicatorsare builtusingdivided differencesanddifferentconstructionscanbefoundintheliterature(seeforexample[13,14]).Thenonlinearweightscon- structedusingthesesmoothnessindicatorsassurethatthestencilsthatcrossadiscontinuityhaveaverysmallcontribution tothefinalapproximation[15–19].

In [20] anewWENO-5 algorithm isdesignedforthecell averages inordertoobtain maximumorderat theintervals closeto discontinuities.In thisworkwe generalizethat algorithm,constructing aWENO-(2r− 1)algorithm foranyr>1, andperformsomeimprovementsorientedtoobtainthesamekindoforderoptimizationclosetodiscontinuitiesforstencils of any length. This papercan be considered as the second partof [1]. In that paper, we presented the new algorithm interpretingthatourdatawasdiscretizedusingthepointvaluesofafunction.

We constructouralgorithmshowinggeneralexplicitformulasforanyr>1.Wedesignnonlinearweights that replace the optimal weights definedin theclassical WENO-(2r− 1) algorithm. Thesenew weights are definedsuch that if there exists a discontinuity,then they tendto a powerofh_l and, in othercase, they tend toa value that allows toobtain the maximumpossibleorderofaccuracy,takingintoaccountallthedatavaluesofthestencilthatareplacedatsmoothzones.

This paper is organizedas follows: Section 2 exposes how the classical WENO algorithm is constructed forthe cell- averagesetting.Section3explainstherelationbetweentheconstructioninthepoint-valuesandthecell-average settings.

Section 4 is dedicated to explain how to construct the new algorithm in the cell averages and to analyze theoretically its accuracy. Section 5 presents some experiments dedicated to analyze numerically the accuracy of the new algorithm throughgridrefinementanalysisclosetodiscontinuities.Wealsopresentexamplesofapplicationofthenewalgorithmto thecompressionofunivariateandbivariatefunctions.Finally,Section6presentstheconclusions.

2. Linearandnonlineartechniques:theclassicalWENOalgorithmforthecell-averages

In thissection weconstruct linearandnonlinearclassical reconstruction operators andtheir corresponding prediction operators(seee.g.[2]formoredetails).Weconsiderafixedvalue0≤ k≤ r− 1andastencilofintervals,

S^r_k

(

^j

)

=

{

^Il_j−r+⁻¹ _k+1,...,I^l_j+⁻¹_k

}

, (6)

withI^l−1_s =[x^l−1_s−1,x^l−1_s ],s=j− r+k+1,...,j+k,andwesupposeapolynomialforeach j,p^r_k,ofdegreer− 1suchthat:

¯f_s^l−1=h⁻¹_l−1

 x^l−1s x^l−1_s−1

p^r_k

(

^x

)

dx, s= j− r+k+1,...,j+k,

thenwedefinethereconstructionoperatoras:

Rl−1

(

^¯fl−1

)(

^x

)

=p^r_k

(

^x

)

.

Finally,usingEq.(5),thepredictionoperatorcanbedefinedas:

(

P_l−1^l ¯f^l−1

)

j=

(

Dl

(

Rl−1

(

^¯fl−1

)))

j=h⁻¹_l

 _x^l

j

x^l_j−1Rl−1

(

^¯fl−1

)(

^x

)

^dx=h⁻¹_l

_x^l

j x^l_j−1

p^r_k

(

^x

)

^dx. (7)

It iseasy tocheck that theprevious definitionprovidesa linearoperator. Asthereconstruction operator isapolynomial, thepreviouspredictionoperatorproducesGibbsphenomenonatthezonesaffectedbydiscontinuities.

Asanexample,ifr=3wehavethreedifferentlinearpredictionoperators:

• Fork=0weobtain:

(

^P_l−1^{l ¯fl−1}

)

2j−1=−1 8 ¯f^l_j⁻¹₋₂+1

2 ¯f^l_j−1⁻¹+5

8 ¯f_j^l−1, j=1,...,2^l−1.

• Fork=1,

(

^P_l−1^{l ¯fl−1}

)

2j−1=1

8 ¯f^l_j⁻¹₋₁+ ¯f^l_j⁻¹−1

8 ¯f^l_j+1⁻¹, j=1,...,2^l−1.

• Andfork=2,

(

P_l−1^l ¯f^l−1

)

2j−1=11 8 ¯f^l_j⁻¹−1

2 ¯f^l_j+1⁻¹+1

8 ¯f_j^l−1₊₂, j=1,...,2^l−1.

Inallthethreecasesandintherestofthepaper,usingEq.(3)we getthat (P_l^l₋₁¯f^l−1)2j=2¯f^l_j⁻¹−(P_l^l₋₁¯f^l−1)2j−1,with j=1,. . .,2^l−1.Then we only need to show the prediction operator forthe cell I^l_2j−1 asthe value ofthe cell I^l_2j can be obtainedthrough(2).Whenthefunctionpresentsadiscontinuity,linearmethodsarenotappropriateduetotheappearance ofnumericaleffectsclosetothediscontinuities.Inthiscase,wecanreplacethelinearinterpolatorytechniqueusedinthe predictionoperatorbyanonlinearmethod,suchasWENOstrategy.NextsectionisdevotedtointroducetheclassicalWENO algorithm.

2.1. TheclassicalWENOalgorithmforthecell-averages

We explaintheWENOalgorithmusingthenotationpresentedin[16]andadaptingittothecell averages.Aswe have presentedinprevioussection,ifweconsiderthestencilS²₀^r−1(^j)=

{

^Il_j−r+1⁻¹ ,...,I^l_j+⁻¹_r−1

}

,wecanconstructthepolynomialp²₀^r−1 ofdegree2r− 2,whichapproximates f inthecellaveragesattheintervalsofthementionedstencil.Wecanalsoconsider thesetofr stencilsofr cellsatthelevell− 1whichcontaintheintervalI^l_j⁻¹=I₂^l_j−1∪I₂^l_j,i.e.S^r_k(^j)^{definedinEq.(6),with} k=0,...,r− 1.Let’sdenoteby p^r_k,withk=0,...,r− 1,ther polynomialsconstructedoverthepreviousstencilsofr cells.

Then, we can combinethesepolynomialsinorder toobtain an interpolationof higherorder.Thus, thereexist valuesC¯_k^r, withk=0,...,r− 1,calledoptimalweights,suchthat



Dl



p²₀^r−1



2j−1=



Dl



r−1 k=0

C¯^r_kp^r_k

2j−1

, with

r−1



k=0

C¯_k^r=1. (8)

In the next section (Section 3,Lemma1) we willgive an explicitexpression forthe optimal weights.As we willsee in Section 4,themainideaofthenewalgorithm isto replacetheoptimalweights,C¯_k^r,bynonlinearweights,such thattheir value dependson the positionof the discontinuity, ifthe discontinuity crossesthe stencil k. Thus, we will consider the followingconvexcombination,

Rl−1

(

^¯fl−1

)(

^x

)

=

r−1



k=0

ω

¯^rk

(

^j

)

^prk

(

^x

)

, (9)

where

ω

¯^r_k(^j)≥ 0,k=0,...,r− 1andr−1

k=0

ω

¯^r_k(^j)=1.Thepredictionwillbedeterminedby:

(

^P_l−1^{l ¯fl−1}

)

2j−1=



Dl

(

^Rl−1

(

^¯fl−1

)) 

2j−1=



Dl



_r

−1 k=0

ω

¯^rk

(

^j

)

^prk

2j−1

. (10)

ClassicalWENOalgorithminthecellaveragesconsistsindesigning thenonlinearweights withtheaimofobtaining order 2r− 1whenthere isnodiscontinuitycrossingthe biggeststencilS²₀^r−1,andoforderr inother case. Inthissense WENO algorithmemulatesthebehaviorofENOalgorithm.In[12]theauthorsproposethealreadyclassicexpression,

ω

¯k^r

(

^j

)

=

α

¯^rk

(

^j

)

_r−1

i=0

α

¯^ri

(

^j

)

^{, k=0,· · · ,r− 1, where}

α

^¯k^r

(

^j

)

= C¯_k^r

( 

+L^r_k

(

j, ¯f

))

^{t, (11)}

wherethevaluesC¯_k^raretheoptimalweightsdefinedinSection3.Theintegert servesthepurposeofmaximizingtheorder ofapproximation.In[13]theauthorssett=2,in[12]andin[1]theauthorssett=r.Inthispaperwetaket=r inorderto verifytheENOpropertyand,atthesametime,maximizetheaccuracy.Theparameter



^{takespositiveandsmallvaluesand}

its purposeistoavoiddivisionsbyzero.Inthisarticlewe set



=10⁻¹⁶.Finally,L^r_k(^j, ¯f)^{arethesmoothnessindicatorsfor} thecell-averagevaluesof f onthestencilS^r_k(^j)^{.In[13]JiangandShuproposed toobtainthesmoothnessindicatorsusing} something similartothetotalvariation,butbasedontheL² norm,sothattheresultissmootherthanthetotalvariation.

InSection3wewillintroducethesmoothnessindicatorspresentedin[21]adaptedtothecellaveragesetting.

Therefore,therearetwoprincipalingredientstodetermineaWENOalgorithm:theoptimalweights,C¯_k^r,k=0,...,r− 1, andthesmoothnessindicatorsL^r_k(^j,¯f)^{.In[1],inordertoobtain}progressiveorderofaccuracyclosetodiscontinuities, the optimalweights arereplaced bynonlinear weightsinthe point-valuediscretization.It isclearthatthere existsa relation betweenthepoint-valueandthecell-averagediscretizations(see[2]).Inthenextsectionwereviewthisrelationandadapt thecomponentsofWENOofthenewalgorithmtothecell-averagediscretization.

3. Therelationbetweenthepoint-valueandthecell-averagediscretizations

Wecananalyzetheproblemofapproximatingusingthecellaveragediscretizationfromthepointofviewofthepoint- valueinterpolationifwedefinetheprimitivefunction:

F

(

^x

)

=

_x

0

f

(

^t

)

^dt, (12)

thatallowstoadapttheWENOalgorithmpresentedin[1]tothecellaveragesettinginanaturalway.

S. Amat, J. Ruiz-Álvarez, C.-W. Shu et al. Applied Mathematics and Computation 403 (2021) 126131

Wedenotethepoint-valuediscretizationasF_j^l:=(D_l^pvF)j=F(^xl_j)^{.Then,settingF}₀^l=0,wecandefinealinearapplication,

l:V^l→V^l,thatallowstogofromthecell-averagediscretizationtothepoint-valuesas:

(

l¯f^l

)

j=h_l

j

i=1

¯f_i^l=

x^l_j 0

f

(

^x

)

^dx=F_j^l, j=1,...,2^l, (13)

anditsinverseas:

(

⁻¹_l ^Fl

)

j=h⁻¹_l

(

^Fj^l− F_j−1^l

)

= ¯f_j^l, j=1,...,2^l. (14) Wesupposethat Il−1(^x; F^l−1)^{isapolynomial}interpolatoroftheprimitiveF(^x),thenwedefinethereconstructionforthe cellaveragesasthederivativeofthereconstructionfortheprimitive,

Rl−1

(

^¯fl−1

)(

^x

)

= d

dxIl−1

(

^x;



l−1

(

^¯fl−1

))

= d

dxIl−1

(

^x; F^l−1

)

. Sincex^l_2j−2=x^l_j−1⁻¹,thenbyEq.(5)

(

P_l^l₋₁¯f^l−1

)

2j−1=

(

Dl◦ Rl−1¯f^l−1

)

2j−1=h⁻¹_l x^l_2j−1

x^l_2j−2Rl−1

(

^¯fl−1

)(

^x

)

^dx

=h⁻¹_l x^l_2j−1 x^l_2j−2

d

dxI_l−1

(

^x; F^l−1

)

^dx

=h⁻¹_l

(

Il−1

(

^xl2j−1; F^l−1

)

− Il−1

(

^xl2j−2; F^l−1

))

=h⁻¹_l

(

^Il−1

(

^xl2j−1; F^l−1

)

^{− F}j^l−1⁻¹

)

=h⁻¹_l

(

Il−1

(

^xl2j−1;

(

l−1¯f^l−1

)

j−1

)

−

(

l−1¯f^l−1

)

j−1

)

.

(15)

The expression in(15) is precisely the formulaof the prediction operator forthe cell average values. Now,we consider the WENO algorithm forthe point-valuediscretization of the primitivefunction F defined in Eq.(12). Letq²₀^r−1 and q^r_k, withk=0,...,r− 1bethepolynomialswhichinterpolateF atthepoints

{

^xlj⁻¹−r,. . .,x^l_j⁻¹_+r−1

}

^and

{

^xlj⁻¹−r+k,...,x^l_j⁻¹_+k

}

^withk= 0,...,r− 1.Nowwecandefine:

I_l−1

(

^xl2j−1; F^l−1

)

:=

r−1



k=0

ω

k^r

(

^j

)

^qrk

(

^xl2j−1

)

, with

r−1



l=0

ω

^rk

(

^j

)

=1, (16)

where

ω

^rk(^j)^{arelinearornonlinearweights.Nowwecanprovethefollowing}proposition:

Proposition1. Let’sconsider f∈L¹(^[0,1])^{andF itsprimitivefunctiondefinedinEq.(12).Letqr}_k,withk=0,...,r− 1bethe polynomialswhichinterpolateF atpoints

{

^xl_j−r+⁻¹ _k,...,x^l_j+⁻¹_k

}

^withk=0,...,r− 1,and

ω

k^r(^j),k=0,...,r− 1,theweightswhich satisfythat,

I_l−1

(

^xl2j−1; F^l−1

)

:=

r−1



k=0

ω

k^r

(

^j

)

^qrk

(

^xl2j−1

)

, with

r−1



l=0

ω

^rk

(

^j

)

=1, (17)

and

F

(

^xl2j−1

)

− Il−1

(

^xl2j−1; F^l−1

)

=O

(

^hsl−1⁺¹

)

,

withr≤ s<2r.Letp^r_kwithk=0,...,r− 1,bethepolynomialswhichapproximate f inthecell-average settingatthestencils S^r_kwith k=0,...,r− 1.Inthiscasewehavethat,

¯f₂^l_j−1−



Dl



r−1 k=0

ω

^rk

(

^j

)

^prk

2j−1

=O

(

^hsl−1

)

Proof. Letq^r_kbetheinterpolatorypolynomialsofdegreer ofF atpoints

{

^xlj⁻¹−r+k,...,x^l_j⁻¹_+k

}

^withk=0,...,r− 1,

q^r_k

(

^xln⁻¹

)

=F_n^l−1, n= j+k− r,...,j+k, (18)

andletp^r_kbethepolynomialsofdegreer− 1whichapproximate f inthecell-averagesatS^r_k(^j),withk=0,...,r− 1,then itiseasytocheckthat:

q^r_k

(

^x

)

=

x x^l−1_j−r+k

p^r_k

(

^t

)

^dt+F_j^l_−r+⁻¹_k, k=0,...,r− 1, (19)

then

I_l−1

(

^xl_2j−1; F^l−1

)

=r−1

k=0

ω

k^r

(

^j

)

^qr_k

(

^xl_2j−1

)

=r−1 k=0

 ω

^rk

(

^j

) 

_xl 2j−1

x^l_j−r+⁻¹_kp^r_k

(

^t

)

^dt+F_j^l_−r+⁻¹_k

= r−1

k=0

 ω

^rk

(

^j

) 

_xl 2j−1

x^l_j−1⁻¹ p^r_k

(

^t

)

^dt+x^l_j−1⁻¹

x^l_j−r+⁻¹_kp^r_k

(

^t

)

^dt+F_j^l_−r+⁻¹_k

= r−1

k=0

 ω

^rk

(

^j

) 

_xl 2j−1

x^l_j−1⁻¹ p^r_k

(

^t

)

^dt+Fj^l−1⁻¹

= r−1

k=0

 ω

^rk

(

^j

) 

_xl 2j−1

x^l_j−1⁻¹ p^r_k

(

^t

)

^dt

+F_j−1^l−1=x^l_2j−1 x^l−1_j−1



r−1

k=0

ω

^rk

(

^j

)

^pr_k

(

^t

)

^dt



+F_j−1^l−1

= h_l



Dl



r−1

k=0

ω

^rk

(

^j

)

^pr_k



2j−1+F_j−1^l−1, thus,



Dl



r−1 k=0

ω

^rk

(

^j

)

^prk

2j−1

=h⁻¹_l

(

I

(

^xl2j−1; F^l−1

)

− F_j−1^l−1

)

, (20)

andbyEq.(14):

¯f₂^l_j−1=h⁻¹_l

(

^F2^lj−1− F₂^l_j−2

)

=h⁻¹_l

(

^F2^lj−1− F_j^l₋₁⁻¹

)

. Thus,by(14),(20)andbyhypothesis,wehave:

¯f₂^l_j−1−



Dl



r−1

k=0

ω

k^r

(

^j

)

^pr_k



2j−1=h⁻¹_l



F₂^l_j−1− F_j−1^l−1− I_l−1

(

^xl_2j−1; F^l−1

)

+F_j−1^l−1



= 2O

(

^hs_l⁺¹₋₁

)

h_l−1 =O

(

^hsl−1

)

.



Thispropositionallows ustodefinethe newWENOmethodusing thealgorithmforthepoint-valuediscretizationde- signed in [1]. The weights

ω

^r_k(^j) ^{are exactly the same in both} discretizations. If

ω

¯_k^r(^j) ^{are the weights defined for the} cell-averagesettingand

ω

^r_k(^j)^{theonesforthe}point-valuesetting,then,

ω

¯k^r

(

^j

)

=

ω

k^r

(

^j

)

, k=0,...,r− 1.

Finally,weonlyhavetodeterminetherelationbetweenthesmoothnessindicatorsforL^r_k(^j,F)^andLr_k(^j,¯f)^. In[20–22]theauthorsproposedtousethesmoothnessindicatorsgivenbytheexpression

L^r_k

(

j,F

)

=

r

i=2

h²_l−1ⁱ⁻¹

 x^l−1_j x^l−1_j−1



dⁱ dxⁱq^r_k

(

^x

)



2

dx, (21)

where q^r_k are the interpolatory polynomials of F at points

{

^xl−1_j−r+k,...,x^l−1_j+k

}

^{with k}=0,...,r− 1. In terms of the cell- averages,theformulawouldbe,

L^r_k

(

^{j, ¯f}

)

⁼

r−1



i=1

h²_l−1ⁱ⁻¹

 _x^l−1

j x^l−1_j−1



dⁱ dxⁱp^r_k

(

^x

)



2

dx, (22)

where p^r_k are the polynomialsthat approximate f inthe cell-averages atthe stencils S^r_k, k=0,...,r− 1. In[20–22] it is proved that thesmoothnessindicatorsinthecell-averages (22)andtheonesforthepoint-valuesetting(21)proposed in [21]arerelatedthroughtheexpression,

L^r_k

(

j,F

)

=h²_l−1L^r_k

(

j, ¯f

)

. (23)

Itiseasytocheckthatthesmoothnessindicatorscanbeobtainedthrough(22)usingthepreviouspolynomials.Theexpres- sionforthesmoothnessindicatorsforr=3cellscanbeexpressedintermsoffinitedifferencesas,

L³₀

(

j, ¯f

)

= 10

3

( δ

²_j−2

)

²+3

δ

²_j−2

δ

j−2+

( δ

j−2

)

², L³₁

(

j, ¯f

)

= 4

3

( δ

²j−1

)

²+

δ

²j−1

δ

j−1+

( δ

j−1

)

², L³₂

(

^j, ¯f

)

= 4

3

( δ

²j

)

²− 3

δ

²j

δ

j+

( δ

j

)

²,

(24)