A Fluid Dynamic Model for T -Junctions

A Fluid Dynamic Model for T -Junctions
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  A FLUID DYNAMIC MODEL FOR  T  -JUNCTIONS ALESSIA MARIGO  AND  BENEDETTO PICCOLI ∗ Abstract.  Motivated by real road junctions, we consider a new fluid dynamic model for trafficflow on networks. In particular at T-junctions, beside some flows distribution and/or merging, therehappen some interactions of cars coming from different roads and going to different destinations.After determining some rules to uniquely solve Riemann problems, we prove existence of solutionson complete networks for initial data with bounded variation (and their limits in  L 1 loc ). Key words.  traffic flow on networks, conservation laws, T-junctions AMS subject classifications.  90B20, 35L65 1. Introduction.  In recent years, many authors contributed to the developmentof fluid dynamic theory for flows on networks, see [4, 6, 7, 10, 11, 12, 13, 14, 15, 18].The most important applications are in urban car traffic [5], telecommunication datanetworks [9, 19, 21], gas pipelines [2], supply chains [1] and others.The main approach used for car traffic is based on the idea of a junction (cor-responding to the nodes of the network), with no particular relationships betweenincoming and outgoing roads. Then, real crossings are modeled splitting them intomany junctions of lower complexity, see e.g. [6, 12]. This means that the traffic flowfrom incoming roads distribute to outgoing roads according to certain preferences,which in [7] were modeled by a traffic distribution matrix. Then, in general, to solveuniquely Riemann problems, i.e. Cauchy problems with constant initial data on eachroad, one also needs to impose the maximization of a functional, e.g. the total flux.In this paper we consider a different point of view for junctions, inspired bymodeling need. To illustrate our approach, let us focus on a simple example of   T  - junction, represented in Figure 1.1. Here we have three roads with both directionsof traffic. Then we can individuate the incoming flows, denoted by 1, 2 and 3, andthe outgoing ones, denoted by  A ,  B  and  C  . Each incoming flux at the junctionsplits into two parts depending on the final destination. Thus flux 1 is split in fluxes1 B  and 1 C   (assuming that  U  -turns are not possible). As Figure 1.1 shows, thereare many interactions among the various fluxes at the junction. However, not allsuch interactions can be considered in the same way. In fact, for instance, fluxes 1 B and 3 B  must flow to the same final direction, thus clearly their sum can not exceedthe possible outgoing flow towards  B . On the contrary, fluxes 1 B  and 3 A  shareconflicting trajectories, but they do not share the same final destination, thus theirsum is bounded only by the junction capacity.To capture this situation, we model the  T  -junction as in Figure 1.2. More precisely,to encompass the whole dynamic happening at the  T  -junction, we use nine virtual junctions denoted by letters  G ,  H  , and  K  . The three junctions  G  are formed by anincoming road and two outgoing roads, thus the described phenomenon is simply aflux split. Such junctions were already modeled in [7].Instead, the three junctions  H  , are formed by two incoming and one outgoing roads.In this case, clearly some right of way or yielding rule should show up to describe thetraffic distribution. This is in fact the case and the theory was first developed in [6].The junctions  K   have a quite different meaning. In this case there are two incoming ∗ Istituto per le Applicazioni del Calcolo “Mauro Picone” - C.N.R., Viale del Policlinico 137, 00161Roma, Italy(  marigo, )1  2  A. MARIGO and B. PICCOLI AC312B1C1B2A2C3B3A Fig. 1.1 .  An example of   T  -junction. AC312BHHH KGGGKK Fig. 1.2 .  The model for a   T  -junction. and two outgoing roads. However, the traffic from each incoming road goes to aprecise outgoing road, while sharing the junction space. Our main aim is then tomodel these new types of junctions. Let us illustrate the mathematical counterpartof this example.We use the Lighthill-Whitham-Richards model ([19, 22]), which consists of a singleconservation law: ρ t  +  f  ( ρ ) x  = 0 ,  (1.1)where  ρ  ∈  [0 ,ρ max ] is the car density,  f  ( ρ ) =  ρv ( ρ ) is the flux with  v ( ρ ) the averagevelocity. A junction  J   is called a crossing junction, if it has the same characteristic of  junctions  K   above. Thus  J   has  n  incoming roads, denoted for simplicity by  I  1 ,...,I  n ,and  n  outgoing roads, denoted by  I  n +1 ,...,I  2 n . Also, we denote with  ρ i ( t,x ) , i  =1 ,...,n , and  ρ j ( t,x ) , j  =  n  + 1 ,..., 2 n , the traffic densities, respectively, on theincoming roads and on the outgoing ones. To describe the dynamics at  J  , we assumethe following: 1)  The flux from road  I  i  is the same of the corresponding exiting road  I  n + i . 2)  The total flux through  J   does not exceed its maximum capacity Γ J  . 3)  The total flux through  J   is maximal respecting rules 1) and 2).In case of high traffic, rules 1), 2) and 3) are not enough to isolate a unique solutionto Riemann problems at  J  . This need of a new rule is not at all surprising, since such  T-JUNCTIONS MODEL  3rules were used both in car traffic for junctions with more incoming than outgoingroads, see [6], and also for telecommunication networks, see [9].Therefore, we introduce a  flux proportion rule  , which is active only when the maximalincoming fluxes overcome the maximal junction capacity Γ J  . More precisely, weassume that there exist ideal equilibrium flux proportions among incoming roads.Thus, there exist coefficients ¯ r i  so that the following holds.FPR) The flux from road  I  i +1  is ¯ r i  times the flux from road  I  i , for  i  = 1 ,...,n − 1.The rule FPR) well captures the situation in the example of the  T  -junction above.Indeed, for instance, the flux 3 A  must yield to the flux 2 C  , thus the correspondingflux proportion coefficient will be less than 1. While, usually the flux 1 B  must giveprecedence to the flux 3 A , unless yielding signs are deciding the contrary.Let us further illustrate the role of rule FPR), for simplicity restricting to the case of two incoming and two outgoing roads. First, FPR) is used only if the sum of incomingmaximal fluxes exceeds Γ J  . Then, to respect FPR), we should set the incoming fluxes γ  1  and  γ  2  so that  γ  2  = ¯ r 1 γ  1 . However, this may be in contrast with 3) (if for example¯ r 1  = 1, the maximal flux  γ  max 1  from road 1 exceeds Γ J   and the maximal flux fromroad 2 is less than Γ J  / 2.) In the latter case, we set the proportion between incomingfluxes so to respect rule 3) (i.e. summing up to Γ J  ) and be as close as possible to thevalue prescribed by FPR).We first show how to define the solution to Riemann problems for the new typeof junctions: the  crossing junctions  . The procedure to define the solution is based onrules 1), 2), 3) and FPR). The obtained solution effectively defines a Riemann solverwith consistency properties, see Proposition 2.4.Then we provide estimates on the total variation of the flux for a wave interacting witha crossing junction, having two incoming and two outgoing roads. Such estimates arethe key point to prove BV estimates on the flux along wave front tracking approximatesolutions.More precisely, a wave front tracking algorithm is defined as in [7], i.e. approximatinginitial data with a piecewise constant function and solving Riemann problems forinteractions between waves and of waves with junctions. To provide a well definedconstruction, estimates on the number of waves and interactions are in order. Thelatter are obtained with a careful analysis based on the special properties of theintroduced Riemann solver. Then, to pass to the limit, BV estimates on the fluxare used, together with standard weak compactness arguments. The final result isexistence of solutions to Cauchy problems on networks.The paper is organized as follows. Section 2 provides the basic definitions andresults from previous papers, while in section 2.1 we describe the Riemann solverfor crossing junctions. The following section 3 contains flux variation estimates forwaves interacting with crossing junctions. Finally, in section 4, we prove existence of solutions on the whole network for  L 1 loc  initial data, which can be approximated byBV functions with uniformly bounded variation. 2. Basic definitions.  We use the same approach as in [15, 7, 11]. For readerconvenience, we recall the main notation and results.We consider a network formed by a collection  I   of unidirectional roads  I  i , mod-eled by real (possibly unbounded) intervals [ a i ,b i ], whose natural order respects thedirection of the road. Roads meet at junctions: each junction  J   is given by a col-lection of incoming roads and outgoing roads, and we indicate by  J   the collection of  junctions. Thus the network can be identified with a directed graph. On each roadthe evolution is given by equation (1.1) and we assume:  4  A. MARIGO and B. PICCOLI (F) The flux  f   is a smooth, strictly concave function (with  f  (0) =  f  ( ρ max ) = 0),thus there exists a unique  σ  ∈  [0 ,ρ max ] such that  f  ′ ( σ ) = 0 and it is themaximum of   f   over [0 ,ρ max ].For notational simplicity, we assume that  ρ max  = 1. Definition 2.1.  We let   τ   : [0 , 1]  →  [0 , 1]  be the map such that   f  ( ρ ) =  f  ( τ  ( ρ )) and   τ  ( ρ )   =  ρ  if   ρ   =  σ . Thus   τ   sends   ρ  to the other density value with the same flux (and   τ  ( σ ) =  σ .)  We restrict to crossing junctions as explained in the Introduction.Then each junction  J   has  n  =  n ( J  ) incoming roads and  n  outgoing roads.Let us fix now a junction  J   and for simplicity assume that the incoming roads are I  1 ,...,I  n  and the outgoing roads are  I  n +1 ,...,I  2 n . A Riemann problem for a system(1.1), on the real line, is a Cauchy problem with Heaviside type initial data. We definea Riemann problem at  J   to be a Cauchy problem with initial datum constant on eachroad. Thus let us fix an initial condition  ρ 0  = ( ρ 1 , 0 ,...,ρ 2 n, 0 ). We look for centeredself-similar solutions (as it is natural for conservation laws see [3]), thus we want todetermine a (2 n )-tuple ˆ ρ  = (ˆ ρ 1 ,...,  ˆ ρ 2 n )  ∈  [0 , 1] 2 n , so that the following holds. Oneach incoming road  I  i ,  i  = 1 ,...,n , the solution consists of the single wave solution tothe Riemann Problem ( ρ i, 0 ,  ˆ ρ i ), while on each outgoing road  I  j ,  j  =  n +1 ,..., 2 n , thesolution consists of the single wave (ˆ ρ j ,ρ j, 0 ).We consider waves with negative speed on incoming roads and positive on outgoingones, thus:ˆ ρ i  ∈   { ρ i, 0 }∪ ] τ  ( ρ i, 0 ) , 1] ,  if 0  ≤  ρ i, 0  ≤  σ, [ σ, 1] ,  if   σ  ≤  ρ i, 0  ≤  1 ,  (2.1) i  = 1 ,...,n,  andˆ ρ j  ∈   [0 ,σ ] ,  if 0  ≤  ρ j, 0  ≤  σ, { ρ j, 0 }∪ [0 ,τ  ( ρ j, 0 )[ ,  if   σ  ≤  ρ j, 0  ≤  1 ,  (2.2)  j  =  n  + 1 ,..., 2 n, . As a consequence, not every flux can be obtained on each road.More precisely, we define: γ  max i  =   f  ( ρ i, 0 ) ,  if   ρ i, 0  ∈  [0 ,σ ] ,f  ( σ ) ,  if   ρ i, 0  ∈  ] σ, 1] , i  = 1 ,...,n,  (2.3)and γ  max j  =   f  ( σ ) ,  if   ρ j, 0  ∈  [0 ,σ ] ,f  ( ρ j, 0 ) ,  if   ρ j, 0  ∈  ] σ, 1] , j  =  n  + 1 ,..., 2 n.  (2.4)The quantities  γ  max i  and  γ  max j  represent the maximum flux that can be obtained bya single wave solution on each road. Remark 1.  We may consider waves with zero speed on incoming or outgoing roads. However, this would generate shocks which stay at the intersection, without entering roads. Since solutions are usually considered in   L 1 , such shocks are not part of the solution on roads (because they affect the value only at one point.) Remark 2.  The maximum fluxes on incoming and outgoing roads can be inter-preted as maximal demands and supplies, according to the theory introduced by J.P.Lebacque, see [16, 17, 11].  Notice that for each road  I  i  and possible flux ˆ γ  i , thereexists a unique ˆ ρ i , satisfying (2.1) or (2.2), such that  f  (ˆ ρ i ) = ˆ γ  i . Moreover, by rule1), outgoing fluxes are obtained once incoming fluxes are fixed. Then we obtain thefollowing:  T-JUNCTIONS MODEL  5 Proposition 2.2.  To solve a Riemann problem at a crossing junction   J  , it is enough to determine the fluxes on incoming roads.  We aim at finding a systematicway of solving RP at junctions as described by next definition. Definition 2.3.  Given a crossing junction   J  , we call Riemann solver for   J  a map  RS   : [0 , 1] n × [0 , 1] n →  [0 , 1] n × [0 , 1] n that associates to Riemann data   ρ 0  =( ρ 1 , 0 ,...,ρ 2 n, 0 )  at   J   a vector   ˆ ρ  = (ˆ ρ 1 ,...,  ˆ ρ 2 n )  so that the solution on an incoming road  I  i ,  i  = 1 ,...,n , is given by the wave   ( ρ i, 0 ,  ˆ ρ i )  and on an outgoing one   I  j ,  j  =  n +1 ,..., 2 n is given by the wave   ( ρ j, 0 ,  ˆ ρ j ) .  We require the consistency condition:(CC)  RS   ( RS   ( ρ 0 )) =  RS   ( ρ 0 ) . Once a Riemann solver is introduced for every junction  J  , we can define theconcept of solution on the network as in [11]. 2.1. Riemann solver at crossing junctions.  The aim of this section is todescribe the solution to a Riemann problem at a crossing junction  J  , using rules 1),2), 3) and FPR).Fix a crossing junction  J   with  n  incoming roads,  I  1 ,...,I  n  and  n  outgoing roads I  n +1 ,...,I  2 n . We denote with  ρ i ( t,x ) , i  = 1 ,...,n  and  ρ j ( t,x ) , j  =  n  + 1 ,..., 2 n the traffic densities, respectively, on the incoming roads and on the outgoing ones andby ( ρ i, 0 ,ρ j, 0 ) the initial data of a Riemann problem. The rules 1), 2) and 3) can berewritten as: 1)  f  (ˆ ρ i ) =  f  (ˆ ρ n + i ) for each  i  = 1 ,...,n , 2)   i  f  (ˆ ρ i )  ≤  Γ J  , 3)   i  f  (ˆ ρ i ) is maximal respecting rules 1) and 2).The rules  1)  and  2)  alone do not give a unique solution (ˆ ρ 1 ,...,  ˆ ρ 2 n ). Moreover,since the solution (ˆ ρ 1 ,...,  ˆ ρ 2 n ) must satisfy the conditions  1)  and  2 ), we denote byΩ the admissible region for the fluxes  γ  i  =  f  ( ρ i )Ω =  { ( γ  1 ,...,γ  n ) :  γ  i  ≤  Γ J  , 0  ≤  γ  i  ≤  γ  ∧ i  ,i  = 1 ,...,n } , where  γ  ∧ i  =  min { γ  maxi  ,γ  maxn + i  } .Let us now quantify rule FPR). We can rewrite the rule as:FPR) ¯ r i  is the ratio among the fluxes on two successive roads  f  (ˆ ρ i ) and  f  (ˆ ρ i +1 ).Now, we want to determine a unique solution to the Riemann problem usingrules 1), 2), 3) and FPR). More precisely, we try to fit rule FPR) as much as possiblerespecting rules 1), 2) and 3).Recall that, by Proposition 2.2, to solve the Riemann problem, it is enough todetermine the fluxes ˆ γ  i  =  f  (ˆ ρ i ) ,i  = 1 ,...,n . (Then ˆ γ  n + i  =  f  (ˆ ρ i ) ,i  = 1 ,...,n .)Then, let us determine ˆ γ  i ,i  = 1 ,...,n . We denote by Γ =   i  γ  ∧ i  . We have todistinguish two cases: I  Γ  ≤  Γ J  , II  Γ  >  Γ J  .In the first case we set ˆ γ  i  =  γ  ∧ i  , i  = 1 ,...,n . Let us analyse the second case in whichwe use the flux proportion coefficients ¯ r 1 ,..., ¯ r n − 1 .Consider the space ( γ  1 ,...,γ  n ) and denote by  γ  r  the point that satisfy the fol-lowing system of equations:   i  γ  i  = Γ J  γ  i +1  = ¯ r i γ  i .  (2.5)Recall that the final fluxes should belong to Ω. We distinguish two cases:a)  γ  r  belongs to Ω,
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