Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving

346 Pages·2013·17.74 MB·English

by  Laurent Gosse (auth.)

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
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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.