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1、JournalofScientificComputing,Vol.28,Nos.2/3,September2006(©2006)DOI:10.1007/s10915-006-9075-yGibbsPhenomena∗PeterD.Lax1ReceivedSeptember30,2005;accepted(inrevisedform)October3,2005;PublishedonlineMay17,2006Inthisnoteweshowthatwhenadiscontinuousinitialvaluepro
2、blemforascalarhyperbolicequationinonespacevariableisapproximatedbyadifferenceschemethatismorethanfirstorderaccurate;itleadstoovershootsanalogoustotheGibbsphenomenonwhendiscontinuousfunctionsareapproximatedbysectionsofFourierseries.AhybridschemeduetoHartenandZ
3、wassremovestheovershoots.Similarphenomenaoccurwhensolvingschemesofhyperbolicequations.KEYWORDS:highorderdifferenceapproximations;Gibbsphenomenon.1.ACCURATEFINITEDIFFERENCESCHEMESWeshalldiscussfirsttheinitialvalueproblemforthemodelequationut+aux=0,u(x,0)=f(x),
4、(1)whereu(x,t)isascalarfunctionoftwovariables,andf(x)isapiece-wisesmoothfunctionwithpossiblediscontinuities,anda(x,t)isasmoothfunction.Forsimplicityweassumethataislessthan1inabsolutevalue.Weapproximatethesolutionofthisinitialvalueprobleminthefor-warddirectio
5、nt>0byexplicittwoleveldifferenceschemesoftheformkvh=cjuh−j;(2)−k∗ToDavidGottlieb,masterofscientificcomputation,subtlenumericalanalyst,Menschextraordinaire.1CourantInstitute,NewYorkUniversity,251,Mercerstreet,NewYork,NY10012-1185.E-mail:lax@CIMS.nyu.edu445088
6、5-7474/06/0900-0445/0©2006SpringerScience+BusinessMedia,Inc.446Laxhereuhrepresentsanapproximationofthevalueofu(hδ,t),andvhtheapproximationofthevalueofu(hδ,t+δ).Consistencyofthedifferencescheme(2)withthedifferentialequation(1)demandsthatcj=1,jcj=a.(3)Asimpl
7、ewaytosatisfytheseconditionsistochoosecF=(1−a)/2,cF=(1+a)/2,(4)−11andalltheothercoefficientszero.Consistencytosecondorderdemandsthat2c22j=1,jc=a,jc=a,(5)plustermsoforderO(δ).AsimplewaytosatisfytheseconditionsistheLWschemecW=(a−a)/2,cW=1−a,cW=(a+a)/2,(6)−10
8、1plustermsO(δ),andallotherc=0.Noticethatwhereasthecoefficientsin(4)arepositiveundertheassumption−1
