Tra c flow modelling with junctions

-

English
22 Pages
Read an excerpt
Gain access to the library to view online
Learn more

Description

Niveau: Supérieur, Doctorat, Bac+8
Tra?c flow modelling with junctions. Magali Mercier ? September 9, 2008 Abstract Motivated by the modelling of a roundabout, we are led to study the tra?c on a road with points of entry and exit. In this note, we would like to describe the modellisation of a junction and solve the Riemann problem for such a model. More precisely, between each point of discontinuity we use a multi-class extension of the LWR model to describe the evolution of the density of the vehicles, the multi-class' approach being used in order to distinguish the vehicles after their origin and destination. Then, we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the di?erent types of vehicles. In the case of the one-T road we obtain a result of existence and uniqueness. This first step allows us to obtain a similar result for the n-T road. We describe these results and also some properties of the obtained solutions, in order to see how long this model is valid. 2000 Mathematics Subject Classification: 35L65, 90B20 Key words and phrases: Hyperbolic Systems of Conservation Laws, Continuum tra?c flow model, Riemann problem. 1 Introduction. Tra?c modelling, in particular from a macroscopic point of view has been intensively investi- gated since the seminal paper by Lighthill & Withham [15] and Richards [17], see for example [16], [10].

  • rc ?

  • let ?1

  • ?1 first

  • tra?c flow

  • bardos-leroux-nédélec ones

  • distributed

  • points periodically

  • continuum tra?c

  • ?1


Subjects

Informations

Published by
Reads 39
Language English
Report a problem


r(t;x)
@ r+@ (rv(r))=0;t x
v
v ‰a
@ ‰ +@ (‰ v(r))=0t a x a

thetsaloforen,trytandthatexit.exitInfact,thisesnote,Univwinniteeewhouldthelikbedel,toedescribypeSeptemtheesses:moratherdellisationtryoftheaherejunctiondelandosolvyedenotedtheout,Riemannaproblemmforehiclessucdistinctionhlaabmothedel.pMorewpreciselyBranze,wbeetroundabwoineenerioeacdherimeter.panointhetpreciselyoftrodisconwtinofuitgoymowofetheuseequation:adellingm(1.1)ulti-classeedextensionratherofthethedierenLtheyWRgoingmowdelthetoalldescribFinallyeytheehicle,evholutioneenofdiscontheMagalidensitDipartimenyStudiofItaliathemercier.magali@ing.unibs.itvseeehicles,Here,thean`mconsiderulti-class'asapproacwithhofbexiteingdistributed,usedpinondsorderout'stoparticular,distinguishnotthetovtracehiclesofafterexit.theirtheorigineandisdestination.bThen,twoinetrytreattractheernedpLoinsotsdensitofvenytrytoanderiesexitethanksatothespMotivecial2008benoundaryw.conditionsethatheregivextensioneWRbinoundstheonthethefromotheywsAsofquitethehadierengivtspt9,yptesvofifvtheehicles.oneInofthevcasewingofentheetone-TwroadtswuiteMercierobtainparacresultdiofsit?existenceBrescia,and25123uniqueness.E-mailThislyrstnetsteporks,allo[11]ws[6].uswtowobtaintatosimilararesultoutforantheroadn-Tproad.tsWeneanddescribpedicallythesesoresultsaanderioalsocorrespsometoproproundabertiespofInthewobtaineddosolutions,wintorderstudytotheseeonhoroadswarrivlongandthisMoremo,delmoiswvinalid.duce2000sucMathematicsthatSubjeetcteenClassicwation:p35L65,ts90B20enKeyandworthedsisandvphrbases:theHypWRerbdel,olicthatSystemstotalofyConservtheationehicles,Labws,tracConstudytinleduumvtractheoarewwmoroundabdel,ofRiemannmoproblem.y1atedInAbstracttrowhereduction.isTgivracspmoladelling,Ininwparticularwillfromconsideraamacroscopiculti-classpofoinLtmoofasview[14],hastiatingbveenafterinplacetensivcomeelyandinplacevareesti-to.gatedthissinceisthearticial,seminalepapvertobeysameLigheedthillw&forWithhamthe[15]ypandofRicehicles.hards,[17],erseeisfordensitexampleof[16],t[10].eThisvpapitereriesonfollooinequationoneacermanenopadress:segmenCamillebUnivwClaudetLyoI,oinbofardtin11y:vroadbreawithjunctions.withF-69622dellingCedex,moranceoTToraryisPmotivtatedInstitutbJordan,yersit?theBernardmoondelling43,ofoulevroundabduouts.noSomeem-pap1918,ersVilleurbannehaFv-eempal-address:readytotacMatematica,kleder-thisdegliproblemdibViay38,consideringBrescia,the-roundabaddroutsmercier@math.univ-ason1.fr,sp1eciala
.
1
2 3
.
‰ ‰1 2
‰3
‰ ‰1 2
‰3
n
0;1v : [0;1]!R C+
r =1
0;1C R+
[0;+1)
q :[r7!rv(r)] q r =r 2]0;1[c c
?: [0;1]7![0;1]
rv(r)=?(r)v(?(r))
?(r)=r r =rc
pteringeedorofexiting.andWeeThewilldenoterst2.treatexistencetheincaseproofthetheb`one-Tasroad'maxim(seeconFigurefor1),structuredandandthendetailsgluewtheofsolutionswobtainedvforoeacathconpinoinist[2],ofoundaryenaretryandandonexitThis(ininorderpreciselytoeobtaineaofloandcaltheingeneraltimeandsolution).hThefolloideaw:areerehiclesehicleswhicrdecreasingv.rbthehitzroadalFigureo1:givAe`T-road'.atisfromthat,eattooneuniquelypenoinequationterofbentimetrytheanderexit,follow2,eecanmodierenetiateprincipalthree3,teyptheestheof`one-TvSectionehicles:givtheofvinehicles2thatmogoresults.straighjunction.t,otheses.ofassumedensitconditionsyeedsecondaryAlltheha,samethewvofehiclesonthatisareconsistabvout[7]:towexitandthesetroad,uousofanddensitinytheof(F)yw,asandconditionstheconcavattainsehiclesbthatChapt.haBardos-Leroux-N?d?lecvseeew,eninspiredteredecialthetreatedroad,uousofbdensitandytscapacitBesides,there-nd.,Thenvw6eoundrequirethethat,ofacrossofthesolution.ppapoinistasofw:disconSectiontinwuitdescribymore,thethedelowwgivofourtoresults;ondingSectioniswconservgived,thetheofoprowinofcasecorrespthetroad';isinless4,thanesomeeprescribdetailsedtheoutputoffunction,theandcase.theDescriptionothewdelsofmainconstrain2.1a`one-T'isGenerallessypthanThroughout,someeprescribtheedwinginputonfunction.spWlae(V)obtaintheinehiclesthisvwtheaspylaaanduniquevwwseaktheen,trophyoundssolution.inIn,orderandtoanishestreattheythe.GoatineHere-Tecase,PwoethehaofvtineLipsconlyfunctionstoR.Colomcollatethegivtervenideassame2factsolutions.atTheeacwhfollopinequalitiesoinentareofoundarydisconstrictlytinvuitandyits,umtheThesenite15].propagation[19,sponeseedtheallo,wingFigureinBelothiswcasedenotetoconditions,obtainbaspuniquethankslothecaltininmaptimedenedwyeakexitentrytropofyoinsolution.theW(1.1).ethecanealsowgivalle6aolosummationwByerthe.local.
q( )
qc
q(r)
(r) rO r 1c .
v(r)=V (1¡r):m
q(r)=V r(1¡r) r =1=2 q =V =4 ?(r)=1¡rm c c m
‰1
‰ ‰2 3
r!v(r)
‰ ‰
@ ‰ +@ (‰ v(‰ +‰ )) = 0 @ ‰ +@ (‰ v(‰ +‰ )) = 0t 1 x 1 1 2 t 1 x 1 1 3x<0; x>0:
@ ‰ +@ (‰ v(‰ +‰ )) = 0 @ ‰ +@ (‰ v(‰ +‰ )) = 0t 2 x 2 1 2 t 3 x 3 1 3
‰ (0;x) = ‰ (x) x2R1 1;0
‰ (0;x) = ‰ (x) x<02 2;0
‰ (0;x) = ‰ (x) x>0;3 3;0
‰ v(‰ +‰ )(t;0¡) = ‰ v(‰ +‰ )(t;0+) ;1 1 2 1 1 3
‰ v(‰ +‰ )(t;0¡) • o(t) ;2 1 2
‰ v(‰ +‰ )(t;0+) • i(t) ;3 1 3
+o i R
‰ ‰ ‰1 2 3
‰1
‰ ‰2 3
‰ ‰ ‰2 1 3
t,articularthatPw.theyeingdensitowdisconokindthebforediagramthetalandundamenconservFw2:theforectivFigureasorout,rareemothemaxdensiteymaximised.ofofthepp,opulation(respthatbneithertheenisterstnorvforotherwise,exitstotherst.road,casethehoforthedensitroad.yoofrst,thebpthatopulationofthathaexits,andconditionstheodensitandyacrossfortsofuitthethep.opulationely(2.3)mandlesstheyfolloroadwingv`bteringoundary'exiting).conditions,pthatecantobyewillcomparedpwithwhictheisonesgoalgivtreatenaine[7],towhereyaehiclestoll-gateonismeansconsidered:maximisethatofendel.ters.onlyAssumingwthatequations,themeansbtheehawsviourwof,drivvers,emoaredelledThesebsignifyythethewsp,eedislaedwthe,oinisofindeptinendenyws:whereaslaomaxofationDescriptionconservectivcaloflo)theusttoedata:thanledcapacitareofinitialsecondaryethiswofdestination,ehiclestheirenand(resporiginelytheirAoththisboinfromwmaxhatetheaddaddprioriterule;witurthermore,notFe(2.2)ossiblecase.decideFhorwamaximisedforOurconcretebexample,towtheeofmaroundabmaxwyctakose.giveprioritandtoInvarethatsomealreadyprescribtheedThisoutputthatandeinputthefunctionswstaking`one-T'vandaluesLetinandthisaftercase,o.ofInthese(2.4)3where‰ ‰1 2
t=0
o;i2BV ([0;+1);[0;q ])c
‰ 2BV (R;[0;1]); ‰ 2BV ((¡1;0);[0;1]); ‰ 2BV ((0;+1);[0;1])1 2 3
(‰ ;‰ ) (t;x)2[0;+1)£(¡1;0)1 2
(‰ ;‰ ) (t;x)2[0;+1)£(0;+1)1 3
x2R ‰ ;‰ ;‰ t=01 2 3
t>0 ‰ ;‰ ;‰ x=01 2 3
' “
2S = ‰=(‰;‰~)2R : ‰‚0; ‰~‚0; ‰+‰~•1
i‚ 0 o‚ 0
¡ ¡‰ (0;x) = ‰ x 2 (¡1;0) ‰ (0;x) = ‰ x 2 (¡1;0)1 21 2
+ +‰ (0;x) = ‰ x 2 [0;+1) ‰ (0;x) = ‰ x 2 (0;+1)1 31 3
o > 0 i > 0 T =f(‰ ;‰ )2S;a•a;b 1 2
‰ +‰ •bg a‚0 b•11 2
S
o2 [";1] " > 0 ‰2 T b < 1 T0;b 0;b
‰ = 0 t = 0 ‰ (t;x) = 0 t x1 1
i· 0 ‰ = 0 ‰ = 0 t = 0 ‰ (t;x) = ‰ (t;x) = 0 t2 3 2 3
x ‰1
¡ +o;‰ ! 0 r ! 1
2
¡‰ ! 01
+o = 0 r = 1
+ ¡o=0 r =1 ‰ >02
¡‰ > 01
¡ ¡‰ = 0 ‰ = 02 1
2.2i.e.wildata,einitialtheonstantsolutionccanwithonandIfoutow,densitiestheLandandinowisthewillforifandforvaluessolutionoundarydensitiesfortrbandonstantWhitham'scearwithept(2.4)whenin(2.3)the(2.5)theIn(respthiselycase,increaseforteractwene,obtainetthethenfollothewingforresult:haveTheoremconserv2.4vUnderasthe[21].hypdisconotheseswher(V)spandr(R)ely,vthecorrespRectiviemannbprleft-sideoblemonehiclesforIncasethenowsproblemmaximisefromobviousbloonkwhereasroadectivsimplynotandthoseparticularRemarkens.triplein(2.2)(2.3)(2.4)theAsenseatofDenitions.thesimultaneDenitionof2.2.timeFnulurthermoree,whilewhenthe(2.2)aanlamee,thewemo(2.2)(2.3)(2.4)osedforo,etherthateuitiesexistswesomecinvariantwhensetinproblemthat,RiemannWBythis2.3folloDenitionroad.canw(whicelotob,denoteturne),Wviour.on(2.4)thesatisfyendinpresenceofofestoacjammedtrinthe,,elya.e.timeforthe,(respforwe4.).densit(2.3)leftsmaltolpriorienoughItand2.1satisfydoininofwithlarwhogeter.enough.2.6OnofesatheisRtoiemannandsolver.forLthetimecDenitiononsider,eous.disprtheseoblemmaximisationis,notatclontinuous.atHowever,allitandis,ctoontinuousisonusualsometosubset:scalarforationacw,trwthereco,era.e.classicalwithWRfordel,3.exp;inandbforok(2.3)Wtoobservsolutionhereysome,tinwithapptropwheneneeakases(cialwalsoaeistheb,eingexcanesolutioninvariant.set),etheexplainsolutionqualitativisasobtainew:dacwhereontinuously.ehiclesQualitativgoejammedpropherties.ondsRemarkthe2.5outIfresp2.ely;lforit(2.2)theatehatimeoftotracsolutiontheyof,roadthendeptropdramaticallyentheeakorwabsenceaviswilling1.gothatthisforroad.allfact,suchtheandbutfunctionsandandectiv(2.2)(2.3)(2.4)ofdecouplesthein),tsamewatoanindepectivendenctthatIBVPs.notFthenromtotaltheytracthepwilloinabruptlyt1;ofifview,aitismeans(respthatelythoseRemarkwhoexitfor)(2.2)thearising(2.3)thec(2.4)edis(2.5)ignoredadmitsnothingahappunique4solutionn
n n 2N n < 1
x 2R k 2 J1;nK x < x < x < :::k k¡1 k k+1
n+1 ‰i;j
i j
2n+1 (n+1) (‰ )i;j i2J0;nK;j2J1;n+1K
.
xx x x x1 i j n
i, j
0 n+ 1
1 i j n .
n
+ ¡x x ‰ = 0 i2J0;nKnfkg x x ‰ = 0k i;k k k;jk k
j2J1;n+1Knfkg
p ii;j
j
(¡1;x ) (x ;+1)1 n
(x ;x ) k2J1;nKk k+1
0 1
X
@ A8i2J0;nK 8j2J1;n+1K; @ ‰ +@ ‰ v( ‰ ) =0:t i;j x i;j l;m
l;m
seepnoteriowdicallyWdistributedadisconetininniteuities.ansWjusteewithhaehiclesvtheeentoaindotroinduceithereinsomeandnotations.wingWthateexitassumelathatt.thetsp,oinwts-Tofalenonetryobtainand,exittoarenitelotocated,inlthetspaddoin(P)tsbdicallyproperioofpWtries(lo,inforweninwithdisconroadexit,innitepancounofcanproblemMore,`withRemarkthedoaddressvehiclestoeeThislikjustould,wthisehappwaloutsucienroundabspa;deliseeformo,to.Fthaturthermore,forwandeennpumWbtheeryptheemainnenroadtrytbofytering0,areandcasethehamainfolloexitconservbonyalsorderconsiderInfact,case.p.ulatingInuitthis,conptext,trywtseercalln-T'with`considertheedensittoynecessaryofcounvloehiclesroad'.whic2.7heennotterlowintoThemorandthanexitturn.inme2.2that5afterFigureto3);sayincasefact,intheensmwhatulti-classforapproaclhtbisyeed,origindestinationpropagationhasthealreadyandbbeeneinthankstrosayeIndeed,(2.6)distributed,thesoprevious,wporkal[14].erioAsexitnotryvofehicleoinendters.inebherewillfolloithorothesis:exitsWinkno0,thewumeersnallyaharepresenvtheeortionout,vroundabenainonthattimestounknoinwns:.theelargervfortheenswinghappcal)whatationstudywstothettervaneasierwrsteew,ifInLater,oindistributed.somedicallyanderioaccum.yptintsofoinforpoinoftheerandb:rofumoinnoftablebcounumatableofacaseroadtheaninalsoentheevconsiderresult,,timegenerallyinout.caseroundabofoftableondserimeterthecalpFigureto3:correspAducedin(seek2J1;nK xk
0 1 0 1
X X
¡ +@ A @ A8i=k;8j =k ‰ v ‰ (t;x ) = ‰ v ‰ (t;x ) max;i;j l;m i;j l;mk k
l;m l;m0 1
X
¡@ A8i2J0;nK; ‰ v ‰ (t;x ) • o (t) max;i;k l;m kk
l;m0 1
X
+@ A8j2J1;n+1K; ‰ v ‰ (t;x ) • i (t) max;k;j l;m kk
l;m
‰i;j
xk
0 1 0 1
X X
¡ +@ A @ A8i=k;8j =k ‰ v ‰ (t;x ) = ‰ v ‰ (t;x ) max;i;j l;m i;j l;mk k
l;m l;m0 1
X X
¡@ A‰ v ‰ (t;x ) • o (t) max;i;k l;m kk
0•i•n l;m0 1
X X
+@ A‰ v ‰ (t;x ) • i (t) max;k;j l;m kk
1•j•n+1 l;m
¡ +x x
k k
+o i Rk k
xk
8 9
< =X
2(n+1)S 2 = ‰2R :8(i;j)2J0;nK£J1;n+1K;‰ ‚0; ‰ •1(n+1) i;j i;j: ;
i;j
S =S2
o ik k
(¡1;x ) (x ;+1) (x ;x ) k2J1;nK1 n k k+1
¡‰ j = ‰ ;i;j i;jt=0;x2(¡1;x )1
k+1=2‰ j = ‰ ; k2J1;nK;i;j t=0;x2(x ;x ) i;jk k+1
+‰ j = ‰ ;i;j t=0;x2(x ;+1) i;jn
T > 0
t2[0;T]
L=minfx ¡x g>k+1 k
k
L
0 T ‚
2V
n=1
andthetherpreviousbsectionarycanofbannounceeidenoftiedthehereinb(2.9)yhypexitingtheandopyengiveteringF.wsThemaximisedRiemann(2.7)problem.eW.enotations,areTheoremin(F)terestedloinoblemwuniqueeaktheene,tropoundyconditionssolutionsvofedthepropproblemall(2.6)b(2.8),owhen6theoundaryfunctionswveehicles:theandthe6e6follosomeUndertin(V)y(P)appexistswhensucheiemannconstantsadmitsandakwhenforwyeFccholowerosetheinitiale:conditionsmakthatinarefunctions,constan,tsomeonQualitativtheAsinbtervinalseingwtheewscouldthein6eing:rstconditionsofbasthenprescribandelomaximiseddenotecase,Wone-T(2.8)theininWithandpreviousAswrule.canandthenotationwing:priorit2.8thethethatothesesthe,,andfor,ofeecausecatedbroadorthatallR,prin(2.6):(2.8)remark(2.9)canaeweWentr.solutionthesecondaryforofw6o.totalurthermorthewetanaccounatobinforetimetakexistencthatletintheconditionswoundaryobaluesstrongertakingconsiderinputtooutputetterthenbprescribisareit.er,eeverties.winHocaseequations.thesey,,disconareuittakphenomenaeneartobeandecorrespinitialondingvto.thecapacitfx• 0;t‚ 0g fx‚ 0;t‚ 0g
‰ ‰1 2

@ ‰ +@ (‰ v(‰ +‰ )) = 0;t 1 x 1 1 2
@ ‰ +@ (‰ v(‰ +‰ )) = 0:t 2 x 2 1 2
¡ ⁄ + ⁄‰ (0;x) = ‰ x2R ; ‰ (0;x) = ‰ x2R ;1 1¡ +1 1
¡ ⁄ + ⁄‰ (0;x) = ‰ x2R ; ‰ (0;x) = ‰ x2R :2 22 ¡ 2 +
¡ ¡ ¡ + + +‰ =(‰ ;‰ );‰ =(‰ ;‰ ) S1 2 1 2
2£2
+ + + + + +r =‰ +‰ s=‰ =‰ ‰ =0 r =‰ +‰ s =‰ =‰1 2 1 2 2 1 2 1 2
‰ =02

0@ r+(v(r)+rv (r))@ r = 0;t x
@ s+v(r)@ s = 0:t x
‰ = 0 ‰ = 0 ‰ =‰ ‰ =‰2 1 2 1 1 2
‰=0 Snf0g
0‚ (‰) = v(r) + rv (r) ‚ (‰) = v(r)1 2
‰ = (‰ ;‰ ) ‚ (‰) < ‚ (‰) ‰ = (‰ ;‰ ) = 0 ‚1 2 1 2 1 2 2? ¶
‰1r = 1 v (‰) = = ‰1 ‰2? ¶
1
v =2 ¡1
‰ = 0 d‚ (‰)¢v (‰) Snf0g q1 1
‰ 2 S d‚ ¢v · 02 2
r s ‰ 2 S
O (‰) = O (‰ ;‰ ) = f(‰ ;‰ ) 2 S;‰ =‰ = ‰ =‰ g ‰ = 01 1 1 2 1 21 2 1 2 2
O (‰)=f(‰ ;‰ )2S;‰ +‰ =‰ +‰ g2 1 2 1 2 1 2
+ + + + + +‰ =‰ =‰ =‰ r •r ‰ = (‰ ;‰ ) ‰ = (‰ ;‰ )1 2 1 21 2 1 2
0 0 +x=t=q (r) x=t=q (r )
incase.and`one-T'eedthealsoforforanalysis.vvhnicaloecremarkT,3studyIfonehicles.areFirst,ew(`straighewandwlareale6hforwThen,in,andw.estraighobtainwsomethinguities.similar[20],b1-wytoconsideringa(3.11)uniqueconsiderethe.classicalfunctionsinsteadvofTheRiemanneasilyproblem,offoreforin,6so'thetheonly[4].problemtheisalsoinshoaesroad1-wwithc.whereasConsequentacttlyw,IfwTeseerstthatsolvstandardeSectiontheentrproblemhinofttherwtheottwith.eThethatcandharacteristicRiemannspts.eedsobtainofa(3.10)esareWypypesehiclesof,vsameehicles,w.ofalsorespSectionectivee,for6densitiesandandplanes,,whicbhofhaandvourandWeparticular,thek-curvsamerarefaction-spandeedlines.lavw.ofWorevfor2-w,arewithconobtainpreciselythehafollofollowingChap.result:systems,Prophositionsee3.1,Lv.fromWypeofremarksystemthatTheetsolution.usrarefactionces,onsiderbthethetwo-pakopulationsexistsfor,systemwill(3.10)vwithwithwhen6theofolsolutionslowingWpiecan2.4.herectheewiseoth3.1smocstrong6indata:arianinitialLet,that,and.thatwonstantvRiemanncurvisarealweaetc.ystnon-negativesev(exceptwhicinhaproblemvwiththetsp).laTheThen,assodenoteciatedeeigen;v3.2,ectorswarewhenw`halfInwhenproRiemannofproblemsof,Theoremandthisquarter.andypFinallyofLetsection,oesColomwBenzoniearticleandonsolv[18]estudythebasedRiemanneproblem13]).forIn(2.2)(2.3)(2.4).theInc.esThistheallocurvwscoincideusaretotseeThethatathees1-cmadeharacteristicshoeldksisrarefactiongenauinelyes,nonlinearthewhenaSectiones6con3.1,dis-wtin,Moreas,eerstvstudythethewing:Riemann12&problem[19,onseeaemplestandardorroadricto:anddo[1];essystems'notthevaanishesingott-linealeneequivtisa,is(3.10)(3.10)bineing3.3,strictlyof.concaPrvopye.areThew2-vcwhicharacteristicareeldetiseenlinearlylinesdegenerateequationssinceweforaallesystemandthew,ewcompleteehao7t+ + + +‰ =‰ =‰ =‰ r >r ‰ ‰1 2 1 2
+ +r v(r )¡rv(r)
c= :
+r ¡r
‰ =(‰ ;‰ )I 1;I 2;I
+r¡ ¡ ¡ ¡s(‰ )=s r(‰ )=r ‰ = ‰ r =0I I I ¡r
.
2
+
I

O rc 1 .
¡ +‰ ‰ 0
¡‰ 0 ‰ =‰ = cst1 2
+¢ + = f‰ 2 S;‰ +‰ = r gr 1 2
+‰
+ ¡q(r )¡q(r ) +c = = v(r )
+ ¡r ¡r
+ +‚ (‰ )=v(r )2
¡ + +‰ = 0 ‰ c = v(r )
¡ +‰ =‰ =0 ‰·0
¡ ¡ +‰ =0 ‰ ‰
+ ¡ +‰ =0 ‰ ‰

S
fx•0;t‚0g fx‚0;t‚0g
x = 0
(3.10)0wtothewhatevdeneerorpisoinvtvintheFiguree(seetoandthethatthesoifstatetotermediateisintranunderndsets,ustanmreconditionwinproblem,ClearlyRiemann,thelinkingea,eandythenya3.22-wowaprvbearfromthethisrin3.2termediatethestatevtoissolvortoe.t.Hoacceptableweevaser,;allhatheseeinbtermediatealongstatesbcorrespcondeinwfactsolutiontomaktheIfsame1-wsolution,.bsetecauseunderthethe1-shoMorccisely,ksezoidsarearoflospalsoeedoworder(3.10).Ingenereedonspeofproblems.ks`half-Riemannccaseshobareproblemtooffrom.gowhenthatwhatNoanesthees.acurvproblemvndaoundary1-ww,thedene,solutionand4).If,andifthee2-wlineawvdeneessolutionareyofasptoeedtoesyv1-shoak;wtendproblem;,Riemann,theeoftheSolutionb4:linkingFigureerw,bsoatheawe.aRemarkvTheeswhenareinvariantattacthehed.ofIfsystemw.eedoenottheacceptapctitiouswhosewoundariesaevHugoniotescibeetinvariantwtheeenoffromsystemr(Fesomevalandesultsainvariantrseb[12]).inHalf-RiemannducedWthecall1-wproblem'.simplegeneralofhainitialeoundarystatesalue[2],in[3]).quartereedplanea6solutions:ortrdierenensobtainhappcouldseeet.wTwhenoinitialsummarise:isifconstanwTheehereliktooksthelobitconditions,w,winSomeattemptcriteriacvtheofeen,trowineliteraturetakanetoonlyharacteriseasetshoattainablec(seek[9],of8sp¡ ¡ ¡ ¡ ¡ ¡ ¡‰ =(‰ ;‰ )2S r =‰ +‰ N(‰ )1 2 1 2
‰b=(‰b ;‰b )2S1 2
8
@ ‰ +@ (‰ v(‰ +‰ ))=0> t 1 x 1 1 2< @ ‰ +@ (‰ v(‰ +‰ ))=0t 2 x 2 1 2‰
¡ ¡> (‰ ;‰ ) x < 01 2: (‰ ;‰ )(0;x)=1 2 (‰b ;‰b ) x > 01 2
r 1c¡ ¡ ¡r ‚r ‰ ‰c ¡ ¡r r
¡?(r ) 1¡ ¡ ¡ ¡r <r ‰ ‰ ‰c ¡ ¡r r
¡‰ 2S min ‰ v(‰ +‰ )=01 1 2
¡N(‰ )
. .
22
C C
E

B

−O r= (r )r 1O c 1 ..
¡ ¡ ¡ ¡ ¡r ‚r N(‰ )=[B;C] r <r N(‰ )=f‰ g[[E;C]c c
¡x = 0 N(‰ )
¡?(r ) 1¡ 0 ¡ ¡ ¡r <r N (‰ )=f‰ g[f‚‰ ;‚2] ; ]gc ¡ ¡r r
b‰ = (‰b ;‰b )1 2
fx=0g
¡rb¡r
¡ ¡ ¡N(‰ )‰f‰2S;‰ =‰ =‰ =‰ g1 2 1 2
¡q(rb)¡q(r )¡rb>r c =
¡rb¡r
¡q(rb)¡q(r )•0
orforwases,kcaoth;Rrighet,ebproblem..Insuchright).negativ5,lineandofeeFiguraeendingLeft-halfthat(se3.3withwne.ointh3.2.1ontainspinthestateswithagetherseento3.1,,agativesp.vRemarka3.4aWhenofwehwantnotrtodenoteconsidereshockssetoftozerifoifspesespetodareinonthealeft-half.prvoblemthe(thatPropartheeesctitioushaasositivtheysoaronlyealoe:ccaterarefactionde,onthetheeaxe,spinee),negativweeed.haveFixtomodifyahalittleshothespsetdThenandpisthe:,Ifnegativinonlytheprc,asewaves:withtheeseeed,gmentorderWknoandwhattakethewithattainableextrtheointsspwemevextrWwithhagmenteseinthepro:ofemeositionpthatoints2-wandv(sealweysxvaplefteIfeed,left);w5,canehaFigurestate1-wandvandeithereshoalkwaewlov.depPronosignof.rTherok.whicaremeanswparticularcases:isequivalentoftoesearcsphingLemmaanrarticialandrighrtstateIfLeft,5:Figure,forethevrighatcrofstateseedattainableby.intheaofproblemointslikthatesolution(3.10)(3.11).theWwhiceisareehereandonlyifiniemannterestedobleminifthecwonlyalvstudyThereofttheoleft-half9problem¡ ¡ ¡† r ‚r q(rb)•q(r ),rb‚rc
¡ ¡ ¡† r <r q(rb)•q(r ),rb‚?(r )c
¡rb•r x=t=
0 ¡ 0q (r ) x=t = q (rb) rb ‚ rc
0 ¡ ¡q r ‚rb‚r r ‚rc c
¡† r ‚r r •rb•1c c
¡ ¡† r •r ?(r )•rb•1c
¡‰
¡‰
¡‰ =0 N(0)=f0g[¢ ¢ =f‰2S;‰ +‰ =1g ⁄1 1 1 2
+ + + ++ + +‰ = (‰ ;‰ ) 2 S r = ‰ +‰ P(‰ )1 3 1 3
‰•=(‰• ;‰• )2S1 3
8
@ ‰ +@ (‰ v(‰ +‰ ))=0> t 1 x 1 1 3<
@ ‰ +@ (‰ v(‰ +‰ ))=0t 3 x 3 1 3‰
> (‰• ;‰• ) x < 01 3: (‰ ;‰ )(0;x)=1 3 + +(‰ ;‰ ) x > 01 3
+r •r T ‰•2S ‰• +‰• •rc r 1 3 cc
+ +r >r T + ‰•2S ‰• +‰• • ?(r )c ?(r ) 1 3
+¢ + =f‰2S;‰• +‰• =r g1 2r
+ +(‰ ;‰ )2S min ‰ v(‰ +‰ )=01 1 31 3
+P(‰ )
+ 0 + + +P r >r P (‰ )=¢ +[f‰2S;?(r )=r >c r
r‚0g
+ +¢ + ‰ +‰ =r P(‰ )1 3r
+‰• ‰
+r•¡r
+r•‚r fx‚0;t‚0g
+r••r r •r••rc c
lines,whIfys.e,aswhenw,c,So,attainablesolutionleftsptheifonadmissiblestatestothe(inhsituationsearcofeewtheandewhenastatethetalwrighonlyawhatevwenoThisxe,eThen,Wcatedifonlyproblem.et-halfTheseRighy3.2.2iemanneof..essenwhenup,yifthis,nowthatesuchif,haisvimpliesetocesontainsvonlysearcwavesbwithypeositiverarefactionspthee.ewdais:planeIfifawwwaavandew:vtheetriangleoblemofthenegativPrwhereideasofofptheointsLemma,replacingeeed'slopositiveFsuchthethateswhenalw.wInwhorderthetoequationcompleteonlydeneecauseeyswFinally,It(sewevFigurhea6,thenleft).2-wIfThatthewprotheof,thatwlinkcaseof:vtheatriangleaecialorspatheendingInof.).esetofephaointsrarefactionnoteetotheequalsummariset,follosuchifthatdenoteconstaneethisbhatovenatakequationsis.solutionttheoandhaensbhappanothingnegativ,sloptoprgetherRwithtothe.lineocase,ThethisofInprostate.arettiallyrighsamearticialforas3.3,thattow`negativstatespthebe`pvehaeed'.ysorareason,alw2-wcanveare(seweaFiguralloeed,6,isright).yInlinebthatothofcesases,andforifalblisFixa3.5inLemmanon-increasing.ha,eed..spalsoethatveositivhapeofsearc,thee1-wevaandrarefactionaddweraave.visayweehestatesofointsnegativcanwhiceeedslopphbwa1-waRemarke.3.61-wAsvbiseforshoe,kwhenawewdovnotdepwantontosigncobtainonsiderparticularshoWcksIfofthezer.o,speevead,wwevhavelotoinchangequarteracanlittlethetheasdenitionw:of,isandiniftheandcease.loincatedcase,beetvwhaintoeeenwthelinesv.10.