Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving PDF
Preview Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving
SIMAI Springer Series 2 Laurent Gosse Computing Qualitatively Correct Approximations of Balance Laws Exponential-Fit, Well-Balanced and Asymptotic-Preserving ABC ToSoniaforallherloveandherpatience SIMAI Springer Series SeriesEditors: NicolaBellomo •LucaFormaggia (Editors-in-Chief) WolfgangBangerth(cid:129)FabioNobile(cid:129)LorenzoPareschi(cid:129)PabloPedregalTercero(cid:129) AndreaTosin(cid:129)JorgeP.Zubelli Volume2 Laurent Gosse Computing Qualitatively Correct Approximations of Balance Laws Exponential-Fit, Well-Balanced and Asymptotic-Preserving LaurentGosse IstitutoperleApplicazionidelCalcolo“MauroPicone”CNR Rome,Italy ISSN:2280-840X ISSN:2280-8418(electronic) SIMAISpringerSeries ISBN978-88-470-2891-3 ISBN978-88-470-2892-0(eBook) DOI10.1007/978-88-470-2892-0 SpringerMilanHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012951183 ©Springer-VerlagItalia2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplicationofthispub- licationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’s location,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissions forusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliableto prosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublica- tion,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrors oromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe materialcontainedherein. 9 8 7 6 5 4 3 2 1 Coverdesign:BeatriceB,Milano Typesetting:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.de) Printing:GrafichePorpora,Segrate(MI) PrintedinItaly Springer-VerlagItaliaS.r.l.,ViaDecembrio28,I-20137Milano Springer-VerlagispartofSpringerScience+BusinessMedia(www.springer.com) Preface Idon’tseeanyproblemwiththemath,butthisisnota dissertationineconomics.Wecan’tgiveyouaPh.D.in economicsforadissertationthatisn’tabouteconomics. It’snoteconomics.It’snotmathematics.It’snotevenbusi- nessadministration. MiltonFriedman,aboutH.Markowitz’smanuscript Balancelawsappearinmanyareasofapplication,rangingfromfluidmechanicsmod- eling,orsemi-classicalWKBapproximationsoflinearquantummodels,todiscrete- ordinate reduction of multi-dimensional kinetic equations. These are partial differ- ential equations describing the evolution in time of intensive (or bulk) quantities which are submitted to a physical process involving both convection and another mechanism(reaction,relaxation,orevendiffusion).Inmanysituations,suchasys- temofequationsstabilizesontoalarge-timebehaviorwhichischaracterizedbyan accuratebalancingbetweenthetransporttermsandtheotherones.Anotherinterest- ingconfigurationistheoneinwhichthesystemcontainsanindependentparameter which variation deeply affects the qualitative behavior of the solutions. We shall therefore speak about qualitatively correct numerical approximations when either (orboth)aforementioneddistinguishedbehaviorscanbereproducedalgorithmically without salient restrictions on the computational grid. Such accurate computations usually result from the use of sophisticate numerical flux functions, which display consistency not only with the convection terms, but with other parts of the equa- tion.Perceivingsimultaneouslyseveral(ifnotall)thetermsappearinginthepartial differentialequationhelpsinpreservingatthenumericalleveldesirablequalitative properties,likedissipationofcertainnorms,respectofpositivelyinvariantdomains, entropyinequalitiesorLyapunovfunctionalsinarobustmanner.Theobjectiveofthe presentbookistoraisethereader’sawarenessofhowsuchelaboratefluxfunctions canbebuilt,mainlyinaone-dimensionalcontextforhyperbolicsystemsadmitting shock-typesolutionsandforkineticequationsinthediscrete-ordinateapproximation aswell.Aneffortwillbededicatedtorigorousmathematicalderivationsandtothe analysisofthenetgainretrievedfromthisapproach. Inparticular,oneshouldoftenkeepinmindthatanequilibriumhastobesought betweenthethreeedgesofthegoldentriangle1:observations,modelingandanaly- sis,numericalsimulation.Whileobservationsareimposedbyoursurroundingworld, modeling canbeinstead achieved atseverallevelsofcomplexity. Amoreintricate 1IlearntthisniceexpressionfromProf.VincentCourtillot. vi Preface model can lead to bigger difficulties in terms of mathematical analysis, even if the development of powerful tools in the field of non-linear analysis allowed to suc- cessfullyresolvedelicateproblemsintermsofexistence,uniquenessandstabilityof appropriateweaksolutions(arousingsomereflexions2aboutwhatiscalledsolving). Impressiveachievements intheoreticalanalysisdon’tyieldautomatically powerful algorithms to simulate efficiently these weak solutions on a computer: concerning balance laws, only Tai-Ping Liu’s extension of James Glimm’s theorem was actu- ally based on an astute numerical algorithm. One insight in that work was a seem- inglysimplefinite-differenceschemewhichbuildingblockcontainsacompletetime- asymptoticwavepattern,includingbothconvectionandsourceterms.Slightlylater, GarySoddevelopedasimilarprocessingforconvection-diffusionsystems,involving againasolverconsistentwithalltheterms.Thereisanunpleasantfactaboutincreas- ing the complexity of a physical model: even if mathematical issues can be over- comebymeansofaneleganttheory,usuallythelevelofnoiseproducedbystandard approximation algorithms increases too. Second-order accurate numerical schemes whichbehavenicelyonsmoothclassicalsolutionscandisplayspuriousoscillations whenaskedtocomputediscontinuouswavesemanatingfrommodelsendowedwith degenerate or vanishing viscosity: the case of the Lax-Wendroff scheme is quite revealing of this type of drawback. Shock solutions are a visual expression of the mathematicalfactthatnostrongdissipationhasbeenkeptattheSobolevlevel:how- ever dissipation helps when designing algorithms because it smears off part of the numericaltruncationerrors.Thegaininaccuracywhenreproducingreal-lifeobser- vationsthatoneobtainsbyincreasingthecomplexityofamathematicalmodelmust alwaysbevastlysuperiortotheincreaseofnumericalnoiseresultingfromdissipation processesbeingremoved.There’slittledoubtthathomogeneoussystemsofconser- vationlawsaresomewhatlimitedwhenitcomestorenderingcertainsituations:when thinkingaboutlarge-scalegasdynamics,gravityisanexternalforcewhichcanhardly bebypassed thusleading totheinclusion ofsourcetermsontheright-handsideof bothmomentumandtotalenergyequations.Suchtermsmakethesystem“lessdis- sipative”, therefore more sensitive to truncation errors as new mechanisms appear likelytoamplifythem.Solversinvolvingawholenon-interacting,time-asymptotic wavepatternsometimescanhelp. Bari,L’AquilaandRome,August2012 LaurentGosse 2ClémentMouhot,Quesignifierésoudreleséquationsdelaphysiquepourunmathématicien? Acknowledgements Ifyouhavebeensuccessful,youdidn’tgetthereonyour own… I am always struck by people who think, well it mustbebecauseIwasjustsosmart.Therearealotof smartpeopleoutthere.ItmustbebecauseIworkedharder thananybodyelse.Letmetellyousomething,therearea wholebunchofhardworkingpeopleoutthere.Ifyou’re successful,somebodyalongthelinegaveyousomehelp. BarackObama,campaigninginRoanoke,Virginia Opportunitiestomakepublicgreetingsarefairlyrare,sothisoneisworthaneffortfor notforgettinganyone.Ibegantobeinterestedinaprofessionalresearchcareerwhen IgraduatedfromUniversityofLille(USTL)backin1992,thankstoateachingassis- tantofstochasticprocesseswhospoketomeaboutthisopportunity.Thingsweredif- ferentatthetimesincesomeonewhodidn’tgothroughtheFrenchCursusHonorum ofGrandesÉcoleswasnonetheless considered abletoattend highlevelcourses,in mycasetheDEA1Analysenon-linéaireAppliquéeatUniversityParis-IXDauphine.I rememberatoughinterviewwithClaudeKipnis(whosadlysuccumbedaheartattack lessthanoneyearlater)whofinallydecidedtogivemeachanceafterwarningmethat Sic’esttropdifficile,n’hésitezsurtoutpasàmecontacteretonvouschangeradeDEA though.Courseswereproceedingatanunusualrhythm,differentfromtheoneheld elsewhere, but teachings by Jean-Pierre Bourguignon, Maria Esteban (2.45 hours forthewrittenexam,nothingmoreandnodocuments)orPierre-LouisLions(Com- bien t’aseu chez Lions?becamearecurrentquestionbeforeChristmas1992)were reallyenlightening.I’vealwaysbeenfondofDifferentialGeometry,butattendingthe course onFinsler metric byPatrick Foulonmade clear tome that Ididn’t have the leveltobeginathesisinthisfield.ThankstoarecommendationbyGrégoireAllaire, I found a financial support to start working on source terms implementation inside aGodunov-typecodeattheFrenchAtomicCommissaryundersupervisionbyboth ImadToumiandPatrickLeTallec.MilitaryServicewasstillmandatoryatthetime henceIhadtointerruptduring1994/95andit’sbeenamatterofpureluckIhadthe authorizationtoleavethebaseinordertoattendthePh.D.defenseofmyfriendAlain Zelmanse:AllaireintroducedmetoAlain-YvesLeRouxwhohadshockedtheaudi- encebysaying,vousmettezuncailloudansunverred’eau,çarendinstablen’importe quelschéma!ThenheinvitedmeinBordeauxandthatwasthebeginningofthe“well- balancedadventure”.Myneighbor,SébastienClercwasverymuchinterestedinthis seemingly new stuff, and there’s been numerous discussions on how to extend the scalarschemetosystems:webasicallydiscoveredthe“non-conservativepath”inde- 1Diplômed’ÉtudesApprofondies,oneyearbeforebeginningaPh.D.thesis. viii Acknowledgements pendentlyandroughlyatthesametime.After35months,itwastimetobegranted the Ph.D., and while asking about Postdoc positions at SISSA to Alberto Bressan who came to visit École Polytechnique, I’ve been answered Do you know Glimm scheme?Alittlebit?Well,notthistime…BenoîtPerthame,whowasrefereeofmy manuscriptsentmetoIACM,inthebeautifullocationofVassilikaVoutoncloseto Heraklionasa2-yearTMRPostdoc,underthesupervisionofbothGeorgiosKosioris andCharalambosMakridakis.IhadthelucktogetmoneytovisitThanosTzavaras severaltimesinMadison,WI,whereIlearntCompensatedCompactness;moreover, ThanosgentlytookthetimetoexplainhowIcouldre-interpretmynon-conservative productswithinthemorerigorousformalismofweak-(cid:2)limits(youstronglyneeda transversalitycondition).Makridakisyouhavetorespectseniority!tookmeintothe development of error estimates for scalar conservation laws, at that time I realized that something was very wrong with the “exponential in time” but I was far from havingenoughskilltocurethisdefect.Evenmorehumiliatingwastheexperienceof being givenbyKosioristhisGeometricOpticsproblemtobestudied intheframe- workofBrenier’sK-multivaluedsolutions:I’vebeenfindingnothingformorethan 2years,andthebestwecouldproducewithmyfriendFrançoisJameswasarigorous analysisofthe…mono-phasesystem!Aftersummer1999IsadlyleftwindyCrete foraone-yearpositionatL’AquilaundertheauspicesofPieroMarcatiwhogaveme manyadvicesontherightmannertosubmitpapers(somethingnobodyevertaught me, back in 1995 Ph.D. students hardly published anything). Life in Italy with the LirawasassweetaslifeinCretewiththeDrachma…Year1999wentbyjustgetting myPh.D/Postdocresearchmaterialacceptedforpublication.Atsomepoint,Igotin touchwithDeboraAmadoriandGrazianoGuerraataCNRconferenceinRome:they werepresentingresultsonBVsolutionsforhyperbolicbalancelawswithdissipation. RememberingtheadvicebyTzavaras,youshouldtrytoprovesomethingforsystems withyourscheme,Iproposedthemtostudytheothercase,wheresourcetermsaren’t sinks, and where transversality (non-resonance) is required. They had the patience to teach me Glimm’s interaction estimates and Bressan’s stability theory despite I probablywasquitedisregardingasastudent.Wefinallycameupwitharathersat- isfying result, that Brenier sold short Quand ça marche avec zéro, ça marche avec uneperturbationd’ordrezéro…PaolaGoatindidafinejobincompletingthisresult bymeansofone-sidedestimatestoo.In2001,Iwasgrantedatemporarypositionat boththeIstitutoperleApplicazionidelCalcoloandUniversitàLaSapienza,inRome whereItaloCapuzzo-DolcettaandMaurizioFalconegavemebacktheK-multibranch problemandRobertoNatalinipushedmeintostudyingtheEuler-Poissonproblem. Suddenly,duringwinter2001,IunderstoodhowtoinitializetheK-momentsystem andrecoveringbothmulti-valuedphasesandintensitiespassedinasplitsecondfrom being ‘infeasible’ to ‘so easy’. On the other side, I realized that Euler-Poisson was atoughproblemofwell-balancingbutIcouldn’tcomeupwithanythinginteresting (Ittookme10yearsmore,despitegivingitatrywithPhilippeBéchouchetoo).At theendofjune2001,Iwasinsomuchdirestraitsthathadn’tIhadcloseathanda job offer from Giuseppe Toscani, I would have probably come back to my parents in Nice and left the profession. Basically all the doors were closed in front of me partly because some people were rumoringthat all my stuff was fake. Fortunately, Acknowledgements ix FrançoisBouchuttookthetimetocheckthedetailsinsomeofmyproofs:Salut,j’ai unequestionsurtondernierpapier:jecomprendsquetuconsiderestoujoursdesflux monotones,puisquedanslelemme7tudemandesengrosu0 positif,etqu’ilyaun principedumaximum,puisqueg(0)=0.Tumeconfirmes?Meanwhile,Toscanitook meintokineticequations,especiallyintheparabolicscaling:itwashotsummer2001 whenweunderstoodthatmostofthewell-balancednon-conservativejumprelations couldberewritteninawaytoproduceaschemenaturallyconsistentwiththelimiting diffusion problem. This led to the nowadays well-known “Gosse-Toscani scheme” andits“magiccoefficient”.Toscanididmore:aftersuggestingmetotakeonawell- balancedschemefortheBoltzmannequation,maiovogliopureunabellaequazione perlatemperatura!,whichIpartlyachievedin2011,heaskedmetogetinvolvedin whathecalled“Wassersteinschemes”,thatnowpeoplecallLagrangianschemesfor diffusion. Again, it was tough to gointo something I knew nothing about … espe- cially that I still didn’t have any permanent position at the time (something people whogottenureveryquicklyneverreallyaccept:‘mobility’istheNewspeakwordfor ‘precarious’andhardlymeans‘excitingadventure’).Atthetime,TMRpostdocswere sufferingincreasingdifficultiesforcomingbackhome,andpersonally,afterseveral yearsoffailure,IhadalreadygivenupapplyinginFrance.Anexplanationmaybethat theseofEuropeanprogramswerecreatingaskilledandversatilehumanoffer,rather used to manage risky projects and sharp deadlines, for which any request fromthe Academicshardlyoccurred!IhadthechancetobeofferedtojointhesezionediBari oftheIACduring2002(5yearsafterPh.D.),justbeforebeinginvitedintheUSAby ShiJintodiscussmymultiphasestuff,everybodycancomputeacuspnowadays,and byAgnèsTourininTorontotoo.Someinterestwasgrowingfortheseuncannyprob- lemsandToscaniplayedagainakeyrolewhenintroducingmetoPeterMarkowich, endof2002.Petergavemeaproblemwhichsoundednothinglessthanimpossibleto me:performingGeometricOpticsfortheSchrödingerequationinacrystalmodeled by an oscillating potential. I pleaded guilty of being totally ignorant about homog- enization, so he gave me a reprint of the famous paper Homogenization limits and Wignertransformswhichdidn’treallytranquilizeme.Letmejustsaythatperforming Blochhomogenizationyieldsafluxfunction(the“energyband”)onedoesn’tknow explicitly: Peter was asking me to do K-multibranch solutions with a flux function nobodyknowswhatitreallylookslike!Ittookmemanyefforts,stimulatedbyallthe Laurent,anynews?e-mailstocomeupwithaworkingalgorithm,whichwastobe extendedtomorecomplexcasesduringtheyearslater,thankstothesupportoffered byNorbertMausertoo,alorsj’aidemandéàYannBrenier,maislestrucsàLaurent, çamarcheoupas?Someconnectionswithamysterious“weakKAMtheory”were evenstressedbyCraigEvans2.Unfortunately,Ifacedanunfaircompetitionagainst moneywhenthefollowingquestionwasraised:howdowedefendourselvesagainst someone who says ‘in 1d, why bother with analytical-numerical homogenization? just overkill the problem with computer power’. There was nothing really to reply exceptthatyoudon’twanttokillaflywithapowerhammer,ordoyou?Atthetime, 2SeeAsurveyofpartialdifferentialequationsmethodsinweakKAMtheory,Comm.PureApplied Math.57(2004)445–480.
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