COUPLING TRAFFIC FLOW NETWORKS TO PEDESTRIAN MOTION. R. Borsche A. Klar

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1 June 4, 7: WSPC/INSTRUCTION FILE pedestrian COUPLING TRAFFIC FLOW NETWORKS TO PEDESTRIAN MOTION R. Borsche A. Klar S. Kühn A. Meurer Department of Mathematics, Universit of Kaiserslautern, P.O.Bo 49, 6765 Kaiserslautern, German. In the present paper scalar macroscopic models for traffic and pedestrian flows are coupled and the resulting sstem is investigated numericall. For the traffic flow the classical Lighthill-Whitham model on a network of roads and for the pedestrian flow the Hughes model are used. These models are coupled via terms in the fundamental diagrams modeling an influence of the traffic and pedestrian flow on the maimal velocities of the corresponding models. Several phsical situations, where pedestrians and cars interact, are investigated. Kewords: traffic, pedestrians, networks, coupling AMS Subject Classification:. Introduction In recent ears a large number of models for traffic and pedestrian flow have appeared on different levels of description. For a recent review on traffic and pedestrian flow models see [5]. On the microscopic level, models which are based on Newton s equations have been developed for traffic flow among man others in [, 7, 4] and for pedestrian flow in [,, 4]. See [5] for further references in both cases. Traffic and pedestrian flow equations on the mesoscopic or kinetic level can be found for eample in [5,, 9,, 4]. Macroscopic traffic and pedestrian flow equations involving equations for densit and mean velocit of the flow are derived in [4,,,, 7, 6, 9] and [, 6]. The classical macroscopic traffic flow model based on scalar continuit equations is described in [7]. Traffic models on networks for this equation can be found among man others in [5, 6, 9,, 8]. In [7, 8, 5, ] pedestrian traffic modeling with scalar conservation laws based on the solution of the eikonal equation have been presented and investigated, see again [5] for further developments and historical comments. Additionall, in [5, 6], a variet of other macroscopic models are discussed. The purpose of the present paper is to develop and investigate a model coupling

2 June 4, 7: WSPC/INSTRUCTION FILE pedestrian pedestrian and traffic flow simulations and describing the interaction of the two tpes of flows. We restrict ourselves to coupling scalar macroscopic models. Traffic flow is considered on a network using a Lighthill-Whitham equation and coupling conditions at the nodes of the network as discussed in [6, 9]. Pedestrian flow is described using the model in [7, 8], where a nonlocal term including a global knowledge of the phsical setting described b the eikonal equation is included into the continuit equation. Both models are coupled together via their flu functions. The state of the vehicular traffic on the one hand influences the pedestrian velocit. On the other hand the fundamental diagram for the traffic is influenced b the pedestrian densit. The numerical methods are based on a first order approach and use a straightforward splitting method to couple the two equations. The paper is organized in the following wa: in section the macroscopic traffic flow model on a network and the pedestrian model are presented. Section contains the coupling conditions for the two models. Section 4 describes the numerical methods and shows numerical results for several different phsical situations including crosswalks and more complicated network situations.. The traffic network model and the optimal path pedestrian model.. The traffic model We consider the scalar Lighthill-Whitham traffic flow model. The car densit = (, t), R, t R +, : R R + R + is governed b the equation t + (f()) = (.) with f() = V () and f : (, m ) R +. Here, f denotes the traffic flu and V is the velocit function for traffic flow with the maimal velocit V () = V m, V ( m ) =. V is chosen such that (V ) = V + V is zero for a value σ (, m ), positive for smaller values of and negative for larger ones. The simplest eample is given b the function V () = V m ( / m ). (.) The traffic network is described in the usual wa: a finite number of roads are represented b intervals [a i, b i ], where a i and b i are the vertices of road e i. We solve t i + f i ( i ) = on ever edge e i = (a i, b i ). To find the junction value i which is derived from ˆ i = i ((b i ) ) for ingoing edges and ˆ i = i ((a i ) + ) for outgoing edges we use the coupling conditions as defined in [9]. The first condition is given b the conservation of flues at the junctions n f( i ) = i= n+m j=n+ f( j ),

3 June 4, 7: WSPC/INSTRUCTION FILE pedestrian where i, i =,..., n are the car densities on incoming roads while j, j = n +,..., n + m are the car densities on outgoing roads. However, these conditions are not sufficient to obtain a unique solution on the network problem. To state further coupling conditions we introduce the following notations as defined in [9]. We denote the distribution rates at the junctions b A := {α ji } j=n+,...,n+m; i=,...,n R m n, such that α ji, n+m j=n+ α ji =, where α ji denotes the distribution rate from edge e i to edge e j. Moreover, we define the sets { [, f(ˆi )], if ˆ i σ Ω i : = i =,..., n, (.) [, f(σ)], if σ ˆ i ma { [, f(σ)], if ˆj σ Ω j : = j = n +,..., n + m, (.4) [, f(ˆ j )], if σ ˆ j ma and define the maimal possible flu on ever incoming road as c i = ma c Ω i c, i =,..., n, and c j, the maimal possible flu on ever outgoing road, respectivel. We restrict from now on to junctions with three roads and use the so called NON-FIFO model [6, ] for three roads. The coupling conditions on i are stated first as conditions on the flues γ i on road e i. In case of one ingoing road e i with i = and two outgoing roads e j with j =, we use γ := min(α c, c ), γ := min(α c, c ), γ := γ + γ. In the situation of two ingoing roads e i with i =, and one outgoing road e we have to consider two possible cases: () c + c > c, i.e. the outgoing road has not enough capacit for the whole incoming flow. Then, ( ( γ := min c, ma c c, c )), ( ( γ := min c, ma c c, c )), γ := γ + γ.

4 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 4 () c + c < c, i.e the outgoing edge has enough capacit for the whole incoming flow. Then, γ := c, γ := c, γ := γ + γ. To find the corresponding densities i we define as in [9] a function τ : [, ma ] [, ma ], τ(σ) = σ, as a map satisfing the following condition τ(), f(τ()) = f(), for each σ. Then, for i {,..., n}, let i [, ma ] such that { { ˆi } (τ( ˆ i ), ma ] if ˆ i σ, f( i ) = γ i, i [σ, ma ] if σ ˆ i ma. For j {n +,..., n + m}, let j [, ma ] be such that { [, σ] if ˆj σ, f( j ) = γ j, j { ˆ j } [, τ( ˆ j )) if σ ˆ j ma. With the above definitions we are able to find a unique solution to the Lighthill- Whitham equations on a network [9]... Pedestrian model We consider the pedestrian model of Hughes [7], where the crowd densit is denoted as = (), R, with : R R +, and the flu function as F : (, m ) R. Then the equation is given b with t + F() = F = UZ, (.5) where U R describes the crowd velocit and Z R, with Z = indicates the desired walking direction. The classical choice for the velocit function is the same as in the traffic case Moreover, the walking direction is given b U() = U m ( / m ). (.6) Z = Φ Φ where Φ, with Φ : R R is determined b solving the eikonal equation U(()) Φ() =. (.7)

5 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 5 Altogether this gives the sstem of equations ( t + U() Φ() ) =. (.8) Φ() combined with (.7). Boundar conditions for Φ are chosen as Φ = on Ω D, where Ω D denotes the pedestrians aim and with Φ = otherwise.. The coupling For the construction of a suitable coupling, we can re-use some ideas of the design of the separate models. We start with the description of the influence of the pedestrians on the flow of cars on the roads... Pedestrian to traffic A road is an object with spatial etension in two space dimensions, but the car traffic is modeled b D equations. In order to match both models, we average the densit of the pedestrians on the road (), (a, b) orthogonal to the driving direction. Suppose the center of the road is a straight line n, (a, b), n R. We define the projection P : R L (R, R + ) R as P () = z/ z/ (n + n )d = (), (.) where z denotes the width of the road. In Figure the averging is sketched for one edge. In order to model the influence of the pedestrian flow on the road traffic, we etend the traffic flu function to f(, ) = g P tot ( )V (), (.) where the rate of driving g P tot is decreasing with increasing pedestrian densit. If there are no people on the street the cars should behave as in the original model, i.e. g P tot is for =, whereas on a full crowded road the cars should not drive at all, i.e. g P tot is for = m. One possible choice is g P tot ( ) = ( / m ) n,. n. (.) The eponent n depends on the situation we want to consider. In front of a school for eample we should use n large to enforce a stronger reduction of speed... Traffic to pedestrian From the perspective of the pedestrians the cars on the road occup some space in their D region. To reflect this we etend the D traffic data onto the domain of the pedestrians.

6 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 6 Prolongation: Suppose that the road is located at n, (a, b), n R. We define the characteristic function for the road i as χi = χ (( n) n) χ( z/,z/) (( n) n ) + χ( z/,) (( a) n) χ( z/,z/) (( n) n + χ(,z/) (( b) n) χ( z/,z/) (( n) n )). The road is not onl stretched to the width z, but also elongated at both ends. This construction is chosen in order to avoid empt edges when two or more roads overlap, as illustrated in Figure. Based on this we define the D car densit of the road i as i () = i ()(χ (( n) n) χ( z/,z/) (( n) n )) + i (a)(χ( z/,) (( a) n) χ( z/,z/) (( n) n )) (.4) + i (b)(χ(,z/) (( b) n) χ( z/,z/) (( n) n )). As data for the etensions at the ends we can use the respective junction values. The averaged D car densit of a network of M roads is thus given b ( PM ˇ () PM i i P, if i χi () > m i χi (.5) () =, else. z z zz Fig. : Enlarged road. Fig. : Enlarging the roads and filling the corners. The influence of the vehicular traffic on the speed of the pedestrians we model b modifing (.6) to U (, ) = gt top ( )U (). (.6)

7 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 7 Accordingl the flu function (.5) is changed to F(, ) = U(, )Z. (.7) For the pedestrian crossing rate g T top [, ] different choices might be reasonable. The first one is similar to (.) g () T top ( ) = ( / m) n, (.8) i.e. the more cars are on the road, the more careful the pedestrians have to cross the street. Again, the eponent n depends on the situation. Another possibilit is to relate the hindrance of the pedestrians to the speed of the cars. This leads to a coupling of the form g () T top ( ) = ( / m) n, (.9) i.e. the faster the cars move, the more complicated is the passage for the pedestrians. Finall we introduce as third option the coupling function ( g () T top ) = ( / m ( ) n ). (.) m Here the pedestrians prefer to cross the street, if there are few but ver fast cars or there are man, but ver slow cars. In all above coupling conditions the eponent n can be used to model the sensitivit of the pedestrians. Higher eponents lead to more destiguished decisions of the pedestrians, whereas smaller values represent a less rigid judgement on the situations. Remark.. If the car traffic or the pedestrian flow is described b two equations for densit and mean velocit ũ, as e.g. in the AW-Rascle equations [], then more sophisticated coupling conditions could be applied. For eample, conditions like g T top ( ) = ( / m )( u/u m ) seem to be more appropriate for the comple decision-making process of the pedestrians. 4. Numerical methods and results In this section we stud the behavior of the above discussed model numericall. Several test cases compare the influence of the choice of the coupling conditions. For the numerical methods we discretize the roads into cells of a constant width h and the D domain of the pedestrians with a quadratic grid of the same spacing. The respectivel averaged densities are denoted b i and ij. 4.. Projections In order to facilitate the discretization of the projections, we consider onl roads aligned with the D grid of the pedestrians.

8 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 8 Pedestrian to Traffic: Consider a situation as depicted in Figure. The indices of the cells orthogonal to the orientation of the road are given b { ij s,..., ij,..., ij+s }. Then formula (.) can be approimated b i = s + ij {ij s,...,ij+s} ij, s = z h where z is the width of the road, h the grid spacing and denotes the floor function., grid edge grid points which overlap the road z s= Fig. : Illustration of the restriction of the pedestrian data onto a D road. Traffic to Pedestrian: Since the roads are discretized such that their grid is aligned with the grid of the pedestrians, we can easil evaluate the epression (.5) for. To ever grid point ij R in the pedestrian domain, we can find all points of the roads which contribute to the discrete prolongated car densit ij. If there is no road closer than z/ to ij, ij is set to. In case that onl one road is nearb, we can just pick the value at the associated road section, as depicted in Figure 6. If several streets are involved or at a junction point, we simpl average over all contributions as in (.5). Here we recall, that due to formula (.4), the junction values can appear in the D domain, although the have no spatial representation in the road network. This enlargement is introduced in order to avoid shorter passages for the pedestrians at junctions, as shown in Figure Numerical methods For the solution of the D conservation law on the network we use a classical Godunov scheme. The D conservation law for pedestrians is solved using the FORCE scheme, see [8]. The eikonal equation is solved b a fast marching method and implemented as discussed in [6]. We solve the coupled problem b a first order splitting method: Algorithm (Coupling procedure)

9 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 9 edge enlarged road grid points z z / z z / z / z / Fig. 4: Enlarging the roads and filling corners. () Project the pedestrian densit onto the network as described in section 4.. () Compute f at the cell interfaces. () Update the traffic network t t + t. (4) Project the new traffic densit onto the pedestrian domain as described in section 4.. (5) Compute the speeds U(, ) according to (.6). (6) Solve the eikonal equation (.7) for the actual time step. (7) Compute the pedestrian flues at the cell interfaces b (.7). (8) Update the pedestrian densities t t + t. 4.. Numerical convergence of the coupling procedure In order to verif the numerical method we stud the numerical convergence on the following test problem. Consider a single road from v = (,.5) to v = (,.5). The maimal densit is m = and the maimal velocit is V m =. As initial condition for the cars we choose init () =.5, and set the boundar condition to bound (t) =.5, t. The road width is z =., the pedestrian domain Ω = [..] [..] and the pedestrian Ω D = [. ] {.} (bottom boundar of Ω). As initial condition for we choose (, ) = {.5, if(, ) [.4.8] [.6], else. The coupling functions are (.8) and (.) with n = n =. The reference solution is computed with a grid spacing of h = 4 and t = 4. For coarser grids the time step grows linearl as h.

10 June 4, 7: WSPC/INSTRUCTION FILE pedestrian h L O L O /.. / / / / / / (a) Numerical convergence of the coupling model Error Pedestrian Traffic 4 Grid size (b) L for different grid sizes h. Fig. 5: Convergence of the numerical method In Figure 5 the errors and the rates of convergence for different grid sizes at time T = are shown. The numerical convergence is close to order in case of the pedestrians and even better for the road traffic. The advantage of the car model, is that the Godunov method is less diffusive than the FORCE scheme and no etra equation as the eikonal equation has to solved Comparison of coupling functions In the following the influence of the choice of the coupling conditions on the overall dnamics of the model is investigated. Therefore we choose a specific test case and compare the results of different coupling functions g T top. We consider an eample of two roads with different maimal velocities. The vertices of the network are located at Domain Ω with network edges.5 Edge v = (, ), v = (, ), v = (, )..5 Edge pedestrians.5.5 Fig. 6: Network plot with pedestrian initial condition. The first edge e = (v, v ) has maimal velocit V m = and m =, whereas the second edge e = (v, v ) has V m = m =. We choose the initial conditions

11 June 4, 7: WSPC/INSTRUCTION FILE pedestrian for the network as,init () =.5.5,,init () =.5 and the boundar condition as,bound (t) =.5.5, t. For the pedestrian domain we set Ω = [..] [..] and choose a road width of z =.. The of the pedestrians id Ω D = [..] {.} {.} [..] (bottom and right boundar of Ω). The pedestrian initial condition is (, ) = {.5, if(, ) [..4] [.6.8], else. In the coupling conditions the eponents n = 4 and n = are used and for the numerical computations the grid size is h =.5 and the time steps are t =.5. We consider three test cases, one for each choice of g (i) T top for i =,,. In Figures 7 to one can see the resulting traffic densit on the left and the pedestrian densit on the right. Traffic densit at time step: Pedestrian densit at time step: Fig. 7: Initial condition for all test cases.

12 June 4, 7: WSPC/INSTRUCTION FILE pedestrian Traffic densit at time step: 5 Pedestrian densit at time step: Fig. 8: Test case : Densities with coupling function g () T top (). Most of the pedestrians choose the first edge to cross the road. Traffic densit at time step: 5 Pedestrian densit at time step: Fig. 9: Test case : Densities with coupling function g () T top (). Most of the pedestrians choose the second edge to cross the road.

13 June 4, 7: WSPC/INSTRUCTION FILE pedestrian Traffic densit at time step: 5 Pedestrian densit at time step: Fig. : Test case : Densities with coupling function g () T top (). One half of the pedestrians takes the first and the other half the second road. The pedestrians are trapped in the upper left corner, so the need to cross one road. Which one depends strongl on the coupling conditions. In Figure 8 we observe that most of the pedestrians choose the road with the smaller densit. In the second case, where the coupling condition depends on the velocit, most of the pedestrians choose the road with smaller maimal velocit but greater densit (compare Figure 9). In Figure it does not matter which road to take since the flues are the same. Thus the pedestrians split up into two groups of equal size, both choosing their shortest path to the. The above eamples show that the solution strongl depends on the choice of the coupling conditions. Therefor a careful selection of the appropriate functions is necessar in order to obtain realistic results Crosswalk An interesting eample to investigate the interaction of pedestrians and cars is a crosswalk. Therefore we consider two vertices

14 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 4. Domain Ω with network edges pedestrians v = (.5), v = (.5), and connect them via one edge: e = (v, v ) Edge Fig. : Network for the crosswalk with pedestrian initial condition. To model the priorit of the pedestrians at the crosswalk on the edge, we choose at the location of the crosswalk a different eponent for the coupling function. For the coupling function on the edge we take n = on the crosswalk and n = 5 otherwise. The maimal densit and maimal velocit are m = V m =. As the initial condition for the network we set init () =.5 and the boundar condition to bound (t) =.5, t. The grid size is h =.5, t =.5 and the road width is z =.. For the pedestrian domain we choose Ω = [.5.5] [.5.5] and set the as Ω D = [..] {.} (left-bottom boundar of Ω). The pedestrian initial condition is (, ) = {.5, if (, ) [.5.] [.8.95], else and coupling function g P tot is g P tot () = ( / m ) 4. For the domain of the crosswalk we set Ω cross = [.4.6] [.45.55]. Now we investigate numericall whether the pedestrians choose the crosswalk (although this wa to their is longer) or if the choose the shorter path where the are more hindered b cars. In the following Figures to 6 the traffic densit on the left and the pedestrian densit on the right are plotted. Initiall the roads have the same initial densit and the pedestrians are located in the upper left corner, Figure. Then the pedestrians walk towards the crosswalk and enter the road on it, Figure. Due to this crossing, congestion in front of the crosswalk arises, Figure 4. In time step 8 the first pedestrians reach the. In Figure 5 we can see a small congestion of pedestrians in front of the. This ist due to the fact that at this point people from top and right want to leave the domain at the corner point of Ω D. When all pedestrians have crossed the road, the congestion in traffic relaes again towards the initial state, Figure 6.

15 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 5 Traffic densit at time step: Pedestrian densit at time step: crosswalk Fig. : Crosswalk at time step : Initial condition of the crosswalk. Traffic densit at time step: Pedestrian densit at time step: crosswalk Fig. : Crosswalk at time step : Pedestrians enter the road on the crosswalk.

16 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 6 Traffic densit at time step: 8 Pedestrian densit at time step: 8 crosswalk Fig. 4: Crosswalk at time step 8: A congestion arises in front of the crosswalk.the first pedestrians reach the. Traffic densit at time step: 6 Pedestrian densit at time step: 6 crosswalk Fig. 5: Crosswalk at time step 6: The congestion has reached the front of road one and most of the pedestrians alread have crossed the street.

17 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 7 Traffic densit at time step: 9 Pedestrian densit at time step: 9 crosswalk Fig. 6: Crosswalk at time step 9: All pedestrians have crossed the road and the congestion wave relaes. In conclusion, as one observes, with the above assumptions, pedestrians choose the crosswalk to cross the road even though this wa is longer Rectangle Finall, we consider a network which forms a rectangle with one ingoing and one outgoing road. The vertices are given b Domain Ω with network edges v = (,.5), v = (,.5), v = (,.5), v 4 = (,.5), v 5 = (,.5), v 6 = (,.5), v 7 = (,.5), v 8 = (,.5) Edge Edge Edge Edge 4 Edge 5 Edge 6 Edge 7 Edge 8 pedestrians Fig. 7: Rectangle network with pedestrian initial condition. and construct eight edges as shown in Figure 7. We use e i for Edge i as short form. The maimal densit and maimal velocit is given b m = V m = on ever edge. As initial conditions for the network we set,init () = 8,init () =.5 and i,init () =.5.5 and i =,..., 7. The eternal boundar condition for the network is,bound (t) =.5, t. As grid size we choose h =.5, t =.5 and

18 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 8 set the road width to z =.. The pedestrian domain is Ω = [..] [..] with the Ω D = [.5.5] {.} (bottom boundar of Ω). As initial condition for the pedestrians we set (, ) = {.5, if (, ) [..8] [.7.9], else. The coupling functions are (.8) with n = and (.) with n = 4. In the beginning it is unclear which wa the pedestrians choose. Either the need to cross the network onl once but with high densit on this road, or the cross the network twice, where the densit on both roads is much smaller. In the following Figures 8 to we can see the traffic densit on the left and the pedestrian densit on the right. At first the traffic is stationar and the pedestrians are located on top (compare Figure 8). In Figure 9 we can see that first pedestrians choose edge e 4 to cross. Hence, in front of this edge congestion in traffic arises and on edge e 6 the densit decreases. In Figure we can observe that most of the pedestrians tr to enter the inner of the rectangle. So the want to cross twice. The congestion on edge e 4 gets higher and there are no cars anmore on edge e 6. Then, most of the pedestrians have entered the inner of the rectangle. Due to the crossing of pedestrians on edge e, e 4 and e 6, densit on edge e 8 is smaller than before. Therefore a few pedestrians choose edge e 8 to cross. The traffic congestion has reached edge e (compare Figure ). In time step 6, the part of edge e 8 where pedestrians cross the street is now at the end of edge eight, since the part with smaller densit is moving with the cars. At t = 6 the first pedestrians reach the (see Figure ). Then the pedestrians leave the road and the traffic congestion relaes again (compare Figure ). Traffic densit at time step: Pedestrian densit at time step: Fig. 8: Rectangle at time step : Initial condition of the rectangle.

19 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 9 Traffic densit at time step: Pedestrian densit at time step: Fig. 9: Rectangle at time step : First pedestrians cross edge four. Traffic densit at time step: 5 Pedestrian densit at time step: Fig. : Rectangle at time step 5: Most of the pedestrians tr to enter the inner of the rectangle.

20 June 4, 7: WSPC/INSTRUCTION FILE pedestrian Traffic densit at time step: 5 Pedestrian densit at time step: Fig. : Rectangle at time step 5: Most of the pedestrians have entered the inner of the rectangle. A few pedestrians choose edge e 8 to cross. Traffic densit at time step: 6 Pedestrian densit at time step: Fig. : Rectangle at time step 6: The part of edge eight where pedestrians cross the street is now at the end of edge eight.

21 June 4, 7: WSPC/INSTRUCTION FILE pedestrian Traffic densit at time step: 5 Pedestrian densit at time step: Fig. : Rectangle at time step 5: Most pedestrians reached the and the traffic congestion relaes. In conclusion, for these parameters most of the pedestrians choose the wa where the have to cross the roads twice. Some pedestrians cross edge e 8 because the traffic densit there was small enough due to crossings of pedestrians on other edges. 5. Concluding Remarks We have developed a coupled model for pedestrian and traffic flow modeling the influence of the two flows on each other. The model is based on scalar macroscopic conservation laws and first order numerics. For the traffic flow we used the Lighthill-Whitham equation on networks and for the pedestrian dnamics Hughes model. There we changed the flu functions b multipling suitable coupling functions which ma differ in special situations. To couple both sstems we defined projections from D to D and vice versa. Then we have shown in different numerical eperiments how the coupled model behaves and what influence the choice of coupling function has. Further research will include the development of higher order coupling and numerical procedures. Moreover, other traffic and pedestrian flow models can be coupled and more detailed coupled models can be developed. References. D. Amadoria, M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Rieman-tpe solutions. Acta Mathematica Scientia,,,, pp A. Aw, A. Klar, T. Materne, and M. Rascle, Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models. SIAM J. Appl. Math. 6 () (), pp A. Aw and M. Rascle, Resurrection of second order models of traffic flow. SIAM J. Appl. Math., 6 (), pp

22 June 4, 7: WSPC/INSTRUCTION FILE pedestrian 4. N. Bellomo, A. Bellouquid, On the modelling of vehicular traffic and crowds b kinetic theor of active particles, Model. Simul. Sci. Eng. Technol., (), pp N. Bellomo, C. Dogbe, On the Modeling of Traffic and Crowds: A Surve of Models, Speculations, and Perspectives. Siam Review 5/, (), pp N.Bellomo, C. Dogbe. On the modelling crowd dnamics from scaling to hperbolic macroscopic models. Math. Models Methods Appl. Sci., 8 (Suppl.) (8), pp M. Bando, K. Hasebe, A. Nakaama, A. Shibata, and Y. Sugiama, Dnamical model of traffic congestion and numerical simulation. Phs. Rev. E, 5 (995), pp R. Bürger, K. Karlsen, J. Towers. A conservation law with discontinuous flu modelling traffic flow with abruptl changing road surface conditions, in: Hperbolic Problems: Theor, Numerics and Applications. Proc. Smpos. Appl. Math., vol. 67, 9, pp G. M. Coclite and M. Garavello and B. Piccoli, Traffic Flow on a Road Network. SIAM J. Math. Anal.,, 5, R. Colombo, M. Garavello, M. Lecureu-Mercier, A Class of Non-Local Models for Pedestrian Traffic. MMMAS (4), 5,. R. Colombo, M. Garavello, M. Lecureu-Mercier, Non -local crowd dnamics. Comptes Rendus Mathematique, 49 (-4), ,. R. M. Colombo, M. D. Rosini, Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 8 () (5), pp F. Berthelin, P. Degond, M. Delitla, M. Rascle, A model for the formation and evolution of traffic jams. Arch. Rat. Mech. Anal. 87, 85-, 8 4. C. Dogbe, On the modelling of crowd dnamics b generalized kinetic models. J.Math.Anal. Appl.87, 5-5, 5. M. Di Francesco, P.A. Markowich, J.F. Pietschmann, M. T. Wolfram, On the Hughes model for pedestrian flow: The one-dimensional case. J. Differential Equations 5 (), pp P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (6), pp J. Greenberg, Etension and amplification of the Aw-Rascle model. SIAM J. Appl. Math., 6 (), pp A. Fuegenschuh, S. Goettlich, M. Hert, A. Klar, A. Martin, A discrete optimization approach to Large Scale Suppl Networks based on Partial Differential Equations. SIAM Scient. Computing,, D. Helbing, Traffic and related self-driven man-particle sstems. Rev. Modern Phs. 7 (4) (), pp D. Helbing, A fluid dnamic model for the movement of pedestrians. Comple Sstems 6, 9-45,99.. D. Helbing and P. Molnar, Social force model for pedestrian dnamics. Phs. Rev. E, 5 (995), pp D. Helbing, I.J. Farkas, P. Molnar, T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations. in: M. Schreckenberg, S.D. Sharma (Eds.), Pedestrian and Evacuation Dnamics, Springer-Verlag, Berlin,, pp M. Hert, A. Klar, Modeling, Simulation and Optimization of Traffic Flow Networks. SIAM Sci. Comp. 5 (), 66-87, 4. S. P. Hoogendoorn and P. H. L. Bov, State-of-the-art of vehicular traffic flow modelling.j. Sst. Control Engrg., 5 (), pp H. Holden and N. Risebro, A mathematical model of traffic flow on a network of unidirectional road. SIAM J.Math.Anal., 4 (995), pp

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