Boundary conditions for first order macroscopic models of vehicular traffic in the presence of tollgates

Applied Mathematics and Computation 234 (2014) 260–266

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Boundary conditions for first order macroscopic models of vehicular traffic in the presence of tollgates M. Dolfin Dep. of Civil, Computer, Construction and Environmental Engineering and of Applied Mathematics (DICIEAMA), University of Messina, Contrada Di Dio (S. Agata), 98166 Messina, Italy

a r t i c l e

i n f o

a b s t r a c t This paper presents a new approach to the modeling of boundary conditions for first order models of vehicular traffic in highways. The first step consists in deriving a model for the dynamics of the flow of vehicles. Simulations of the parameters lead to a detailed analysis of the qualitative properties of the model. Subsequently, for such model, the statement of initial-boundary value problems is deduced, with domain decomposition, for a tract of highway between tollgates. Ó 2014 Elsevier Inc. All rights reserved.

Keywords: Traffic flow Macro-models Nonlinearity Flow dynamics

1. Introduction The modeling of vehicular traffic, as known [13,2], can be developed at different representation scales, namely the microscale, where the dynamic of all driver-vehicle subsystems is individually considered, and the macro-scale, suitable to provide the time and space evolution of the macroscopic flow quantities, typically local density and mean velocity. An intermediate link between these two scales is offered by the kinetic theory approach, where the dependent variable is a probability distribution over the micro-scale state of the interacting vehicles. The critical analysis proposed in [2] shows that none of the aforesaid scale is fully satisfactory in capturing the complexity of the system under consideration. Possibly multiscale approaches need to be developed. More in details, models at the macro-scale are not fully consistent with the physics of the granular, intermittent, flow of vehicles. However, they can offer models with a simple structure useful for the applications. First order models [7,1] simply consist in the mass conservation equation properly closed by a phenomenological model referring the local mean velocity to the local density conditions, including, in some general cases, density gradients; nonlocal effects may also be considered [14]. The book by Kerner [16] illustrates how empirical data can be properly collected and interpreted with the attempt of breaking the complexity of the system under consideration. As an example, in some recent papers [18–21] car-following models considering the relationship between micro and macro variables and taking into consideration proper anticipation effects are considered. The formal structure of first order models is as follows:

@ t q þ @ x ðq B½qÞ ¼ 0;

with

v ¼ B½q;

ð1:1Þ

where B, which approximate the mean local velocity, is an operator to be properly determined according to models suitable to take into account the dynamics at the micro-scale, and the square brackets are used to denote that B is a functional of q; in simple cases, such as those treated in the following, it is a function of q and its space derivative. E-mail address: mdolfi[email protected] http://dx.doi.org/10.1016/j.amc.2014.02.038 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

M. Dolfin / Applied Mathematics and Computation 234 (2014) 260–266

261

Moreover, each vehicle is modeled as a point, i.e. its length is negligible with respect to the length of the road, although a maximal density nM corresponding to bumper-to-bumper packing situation is considered; q ¼ n=nM , is the dimensionless number density of vehicles, where the number density of vehicles n is referred to the maximal density nM ; v ¼ V=V L , is the mean velocity referred to the limit velocity V L , namely the maximal velocity that a vehicle can technically reach in a certain road. This quantity is related to V M (maximal mean velocity of vehicles in free flow conditions) by the relation V L ¼ ð1 þ lÞV M where l is a positive constant valid in all environments (0 < l < 1). These quantities depend on time and space, namely q ¼ qðt; xÞ and v ¼ v ðt; xÞ, where t is a dimensionless time being referred to ‘=V M , being ‘ the length of the road and x the dimensionless space being referred to ‘. Finally, local dimensionless flow is obtained by the relation qðt; xÞ ¼ qðt; xÞ v ðt; xÞ. These models can be implemented for applications with special attention to networks [6,15,8], where the computational simplicity of first order models is a basic requirement to deal with the complexity of large systems. However, the modeling approach should take into account the following: 1. The closure v ¼ B½q should be related to the dynamics at the lower scale and to the quality of the road-environment conditions; 2. The application of the model is effective if suitable boundary conditions can be implemented corresponding to real traffic conditions such as tollgates, junctions, and traffic highlights. The mathematical model proposed in [11] is specifically focused on the first issue to derive a model for the time–space dynamics of the density. The derivation includes a phenomenological model referring the local mean velocity to the local density and density gradient. However it is useful looking ahead to applications and to the validation of models based on empirical data. Accordingly, this paper derives from [11] a model for the flow and focuses on the implementation of boundary conditions corresponding to the presence of tollgates. The approach can be technically generalized to junctions and other devices of networks. More precisely, Section 2 deals with the derivation from Eq. (1.1) of a model, where the flow, rather than the density, is the dependent variable. Section 3 presents the implementation of boundary conditions by a detailed analysis of the modification of the flow dynamics due to the presence of tollgates. Finally, Section 4 proposes a critical analysis towards further developments. 2. Derivation and qualitative properties of a model for the flow As already mentioned in Section 1, it is convenient replacing the equation for the density by an equivalent model for the flow q ¼ qðt; xÞ. In fact, this is a quantity of greater interest for the interpretation of the flow conditions due to the fact that empirical data for the flow are more accurate than those for the density. Moreover, the implementation of boundary conditions can be viewed, as we shall see, as an external action to control the flow of vehicles at tollgates rather than to control the density. Let us briefly summarize the mathematical model proposed in [11] focusing on the case of negligible free flow conditions, namely when the quality of the road and of the environmental conditions are such that the velocity of vehicles reduces even at small densities. The said model can be written as follows:

~ Þ @ x q ¼ K  ðq; @ x q; a ~ Þ @ xx q; @ t q þ H ðq; @ x q; a

ð2:1Þ

~ < 1 is a parameter which takes into account the quality of the road-environment. More precisely a ~ ¼ 0 correwhere 0 6 a ~ ¼ 1 to the best ‘‘ideal’’ ones that might even not be reached in practical cases. Here and sponds to the worse conditions and a in the sequel, the superscript ‘‘’’ is used as a compact form to indicate the two different cases of positive and negative density gradient; in fact the concept of perceived (or apparent) density (introduced in [9]) has been used in [11], by the introduction of local gradients of the density:

qþa ðq; @ x qÞ ¼ q þ ð1  qÞ tanh2 ð@ x qÞ; qa ðq; @ x qÞ ¼ q  q tanh2 ð@ x qÞ;

ð2:2Þ

for positive and negative density gradient respectively, such that a closure condition of Eq. (1.1) is derived as the following ~: expressions which depend on the local density, its gradient, and on aforesaid parameter a

v  ðq; @ x q; a~ Þ ¼

a~ a~ þ ð1  a~ Þ exp



2

ðq a ðq;@ x qÞÞ 1q a ðq;@ x qÞ

ð2:3Þ

:

The explicit expressions of the coefficients of Eq. (2.1) are as follows:

~ Þ ¼ v  ðq; @ x q; a ~Þ q H ðq; @ x q; a

~Þ 1  v  ðq; @ x q; a 2

cosh ð@ x qÞ

! M ðq; @ x qÞ þ 1 ;

ð2:4Þ

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M. Dolfin / Applied Mathematics and Computation 234 (2014) 260–266

and

~ Þ ¼ 2qðq  1ÞM þ ðq; @ x qÞ v þ ðq; @ x q; a ~ Þð1  v þ ðq; @ x q; a ~ ÞÞ K þ ðq; @ x q; a

tanhð@ x qÞ 2

cosh ð@ x qÞ

ð2:5Þ

;

in case of positive density gradient, while

~ Þ ¼ 2q2 M  ðq; @ x qÞ v  ðq; @ x q; a ~ Þð1  v  ðq; @ x q; a ~ ÞÞ K  ðq; @ x q; a

tanhð@ x qÞ 2

cosh ð@ x qÞ

ð2:6Þ

;

holds in case of negative density gradient. Moreover the following auxiliary term 2

M ðq; @ x qÞ ¼

ð1  qa ðq; @ x qÞÞ  1

ð2:7Þ

ð1  qa ðq; @ x qÞÞ2

has been introduced. Let us now deal with the derivation of a mathematical model for the flow q ¼ q ðt; xÞ (for the two cases of positive and negative density gradient) from the model for the density above summarized. Accordingly, let us introduce the vector varT iable z ¼ fq ; q; wg , where the variable w is defined as follows: w ¼ @ x q. The result is in the following: Proposition. The model stated in Eq. (2.1), corresponding to the mass conservation equation (1.1), is equivalent to the following system of PDEs, which define the time and space dynamics of the variable z :

8  > < @t q @t q > : @t w

~ Þ@ x q þ B ðq ; q; w; a ~ Þ@ xx q ; ¼ A ðq ; q; w; a ¼ @ x q ;

ð2:8Þ

¼ @ xx q :

where

~ Þ ¼ v  ðq; w; a ~Þ q A ðz ; a

~Þ 1  v  ðq; w; a 2

cosh ðwÞ

! M ðq; wÞ þ 1

ð2:9Þ

and

~ Þ ¼ 2qþ ð1  qÞ Bþ ðzþ ; a

~Þ 1  v þ ðq; w; a 2

cosh ðwÞ

M þ ðq; wÞ tanhðwÞ;

ð2:10Þ

for positive density gradients, while

~ Þ ¼ 2q q B ðz ; a

~Þ 1  v  ðq; w; a 2

cosh ðwÞ

M  ðq; wÞ tanhðwÞ

ð2:11Þ

holds for negative density gradients. Proof. Eq. (2.8)2 simply expresses mass conservation, while (2.8)3 follows from the definition of the variable w, i.e.

@ t w ¼ @ t @ x q ¼ @ x @ t q ¼ @ xx q : Eq. (2.8)1 follows directly from the explicit expression of the evolution equation for the density in the original model. In ~ Þ, using also the closure condition (2.3), and mass conservation (2.8)2, fact by derivation of the relation q ¼ qv  ðq; w; a yields the formal expressions of the evolution equations for flow and velocity respectively as follows:

~ Þ@ x q þ q@ t v  ðq; w; a ~Þ @ t q ¼ v  ðq; w; a

ð2:12Þ

~ Þ ¼ @ ðq Þ v  ðqa ðq; wÞ; a ~ Þð@ q qa @ x q þ @ w qa @ xx q Þ; @ t v  ðq; w; a a

ð2:13Þ

and

recalling that the expressions for the perceived density in the new variables are obtained simply by replacing @ x q with w into Eq. (2.2) and into the position (2.7). Straightforward calculations lead to the following explicit expression for the first term on the right hand side of Eq. (2.13)

~ Þ ¼ ð1  v  ðq; w; aÞÞM ðq; wÞ v  ðq; w; a ~ Þ: @ ðqa Þ v  ðqa ðq; wÞ; a

ð2:14Þ

Moreover,

@ q qa ðq; wÞ ¼

1 2

cosh ðwÞ

;

@ w qþa ðq; wÞ ¼ 2ð1  qÞ

tanhðwÞ 2

cosh ðwÞ

;

@ w qa ðq; wÞ ¼ 2q

tanhðwÞ 2

cosh ðwÞ

:

ð2:15Þ

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M. Dolfin / Applied Mathematics and Computation 234 (2014) 260–266

The explicit expression of the evolution equation for the flow in case of positive density gradient is obtained by casting Eqs. (2.14), (2.15)1 and (2.15)2 into (2.12), that yields þ

þ

@t q ¼  q

~Þ 1  v þ ðq; w; a 2

cosh ðwÞ

!

þ

~ Þ @ x qþ þ 2ð1  qÞ qþ M ðq; wÞ þ v ðq; w; a þ

~Þ 1  v þ ðq; w; a 2

cosh ðwÞ

M þ ðq; wÞ tanhðwÞ @ xx qþ ð2:16Þ

which is the evolution equation for the flow (2.8)1 with the positions (2.9) (for positive gradient) and (2.10). The explicit expression of the evolution equation for the flow in case of negative density gradient is obtained by using Eq. (2.14) together with Eqs. (2.15)1 and (2.15)3, into (2.12), yielding

@ t q ¼  q

~Þ 1  v  ðq; w; a 2

cosh ðwÞ

!

~ Þ @ x q þ 2q q M  ðq; wÞ þ v  ðq; w; a

~Þ 1  v  ðq; w; a 2

cosh ðwÞ

which is again Eq. (2.8)1 with the positions (2.9) (for negative gradient) and (2.11).

M  ðq; wÞ tanhðwÞ @ xx q

ð2:17Þ

h

Remark 2.1. Eq. (2.8) writes in compact form as the following quasi-linear homogeneous system of PDEs of the second order

~ Þ@ x q þ B ðz ; a ~ Þ@ xx q ; @ t z ¼ A ðz ; a

ð2:18Þ

where the two vector functions T

~ Þ ¼ fA ðz; a ~ Þ; 1; 0g ; A ðz; a

T

~ Þ ¼ fB ðz; a ~ Þ; 0; 1g ; B ðz; a

ð2:19Þ

have been introduced. The second-order term of Eq. (2.18) describes a diffusive process, while the first-order (in space) term describes a convective (or advection) process; in the analyzed situation no zero-order term, representing sources or sinks, is present. Therefore, model (2.18) depicts a nonlinear convection–diffusion process. ~ Þ encodes a tendency of the transport of the flow (positive or negative Remark 2.2. The nonlinear coefficient function A ðz; a ~ Þ will be called in the following ‘‘flow convection rates’’. The transport of the flow is in convection); the quantities A ðz; a the positive direction of the space coordinate when the flow convection rate is positive or in the negative direction of the space coordinate when it is negative (see [22] for linear and quasi-linear convection equations related to other types of physical systems.) Remark 2.3. If @ x q ¼ 0 then flow convection alone is present and the following convection equation for the flow is obtained

2

0

@ t q ¼ 4q@1 

1

3

a~ q2  2q A a~ 5 @ x q:  2 q2 q2 ~ ~ ~ ~ a þ ð1  aÞ expð1qÞ ðq  1Þ a þ ð1  aÞ expð1 qÞ

ð2:20Þ

A detailed analysis of the properties of the flow convection rates can be obtained by the aid of some computing to over~ ¼ 0:5 and come the difficulty of the nonlinearity that characterize the said terms. Detailed calculations are presented for a consist in plotting the flow convection rates and the diffusion coefficients versus density and density gradients. The results are displayed at the end of the paper. The analysis of the plots (see Figs. 1 and 2) induces the following observations:  One can notice the presence of a threshold function relating q and @ x q (specifically put in evidence in Fig.1b), such that below it only forward flow transport is depicted; above it backward flow transport or no flow transport at all is present.  In case of positive density gradient, a forward flow transport is present only for small values of the density when large density gradient values are present; otherwise, a forward flow transport is admissible in correspondence of bigger, with respect to before, values of the density, when the density gradient values are smaller; moreover a threshold value for the density gradient is depicted such that above it no flow transport is present, whatever value of the density is considered.  In case of negative density gradient, a forward flow transport is admissible in correspondence of bigger values of the density; moreover a threshold value for the density gradient is depicted such that below it only forward flow transport is present, whatever value of the density is considered.  Three regions are depicted characterizing forward flow transport (red zone), backward flow transport (blue zone) and no flow transport (yellow zone). Let us now consider the interpretation of the flow diffusion phenomena according to Fig. 2, where the diffusion coefficients are plotted. As before, a qualitative physical behavior is put in evidence by the above plots:  The effect of traffic diffusing down the flow gradient can be deduce by analyzing the signs of the diffusion coefficient func~ Þ for positive and B ðz; a ~ Þ for negative density gradient respectively, which can be visualized by plotting them tions Bþ ðz; a versus density and density gradient as in Fig. 2.

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M. Dolfin / Applied Mathematics and Computation 234 (2014) 260–266

(a)

(b)

flow convection rate

−0.5

−1 0 −1.5

0.2

2 1.5

0.4

−2 0

1

2

0.2

0.6

1

0.4

0.5 0

0

0.6

−0.5

0.8

−1

−1

0.8 1

−2

∂ρ

ρ

−1.5

ρ

1

−2

∂ρ

Fig. 1. (a) Flow convection rate and (b) threshold function.

diffusion coefficient

0.5 0 −0.5 −1 −1.5 0 0.2

2 0.4

1 0.6

0 0.8

−1 1

−2

ρ

∂ρ

Fig. 2. Diffusion coefficient.

 The signs of the diffusion coefficients are concordant with the density signs. The traffic flow ‘‘expands’’ from regions with higher value of traffic flow to regions with lower values and it is driven, in the present model, by flow gradients of vehicles; this effect is determined by density gradient inducing a psychological effect on drivers who ‘‘feel’’ a local density greater than the real one when the gradient density is positive and the opposite when the gradient density is negative and this phenomenon reflects on flow gradients due to the fundamental relation q ¼ qv . Then, the diffusion term of Eqs. (2.18) is generated by a modeling of the adaptation of the driver to local density gradient and it is not artificially plugged into the model (for a detailed discussion on this point see [9]). The effect of diffusion is of smoothing down the flow transport phenomenon. 3. Modeling the dynamics near tollgates and boundary conditions The solution of initial-boundary value problems needs the implementation of boundary conditions that, in the presence of tollgates, cannot be obtained simply by adding the flow conditions at x ¼ 0 and x ¼ 1. In fact, the dynamics presented in the preceding section is modified, in the tracts of road near tollgates, by panels that command a reduction of velocity as well as by a natural trend to reach the velocity in the intermediate tract. Namely vehicles are induced to decelerate when approaching to the tollgate or they are naturally induced to accelerate when leaving it. Therefore, the modeling of the dynamics needs to be modified by taking into account the aforesaid specific features. These will be called ‘‘free dynamics’’ and ‘‘constrained dynamics’’, respectively.

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The approach proposed in the following refer to System (2.8) and it is based on the following assumptions:  The flow of vehicles in the tollgates can be measured at x ¼ 0 and x ¼ 1 as known functions of time:

q0 ðtÞ ¼ qðt; x ¼ 0Þ;

and q1 ðtÞ ¼ qðt; x ¼ 1Þ;

where the intermittent behavior of the flow is regularized by interpolation over time of the data at discrete times. In the simplest case, these two flows are constant with respect to time.  The space domain D ¼ ½0; 1 is decomposed into three sub-domains: D ¼ D1 [ D2 [ D3 , where D1 ¼ ½0; d1  D2 ¼ ½d1 ; 1  d2 , and D3 ½1  d2 ; 1. The dynamics in D2 is delivered by Eq. (2.8), while the dynamics in D1 and D3 needs to be modified by the presence of tollgates.  The dynamics in D1 and D3 can be modified by a superposition of two effects, namely that corresponding to (2.8) (‘‘free dynamics’’), which however vanishes at x ¼ 0 and x ¼ 1, and that induced by the tollgates (‘‘constrained dynamics’’), which vanishes at x ¼ d1 and x ¼ 1  d2 . Let us now develop the modeling of the dynamics in D1 ; D2 and D3 according to the assumptions in the third item. Subsequently the overall statement of the initial-boundary value problem is given consistently with the domain decomposition stated in the third item. Bearing all this in mind, let us consider the following smooth weight functions:

(

u1 ðx 2 D1 Þ ¼ 1  exp 

)

x2 ðd1  xÞ

u1 ðx 2 D2 [ D3 Þ ¼ 0;

;

2

u2 ðx 2 D2 Þ ¼ 1; u2 ðx 2 D1 Þ ¼ u2 ðx 2 D3 Þ ¼ 0

ð3:1Þ ð3:2Þ

and

(

u3 ðx 2 D3 Þ ¼  exp 

)

ð1  xÞ2 2

ð1  d2  xÞ

;

u2 ðx 2 D1 [ D2 Þ ¼ 0;

ð3:3Þ

such that u1 ðx ¼ 0Þ ¼ 0; u1 ðx ¼ d1 Þ ¼ 1; u2 ðx ¼ 1  d2 Þ ¼ 1; u2 ðx ¼ 1Þ ¼ 0. Moreover, their derivatives are equal to zero at the boundary of their respective domains. These functions can be used to weight the two effects mentioned above, namely the trend to the free and constrained dynamics. The corresponding model for (2.8) is as follows:

~ Þ@ x q þ uðxÞ B ðz; a ~ Þ@ xx q þ Cðq; tÞ; @ t z ¼ uðxÞ A ðz; a

ð3:4Þ

with uðxÞ ¼ ð1  u1 ðxÞÞ þ u2 ðxÞ þ ð1  u3 ðxÞÞ. Moreover, Cðt; q; xÞ ¼ fC 1 ðt; q; xÞ; 0; 0gT , with

C 1 ðt; q; xÞ ¼ e u1 ðxÞðq0 ðtÞ  qÞ þ e u2 ðxÞðq1 ðtÞ  qÞ;

ð3:5Þ

where e is a positive constant modeling the free and constrained dynamics near tollgates. Therefore the dynamics of the system under consideration can be obtained by solution of the initial-boundary value problems for Eq. (3.4), where boundary conditions are delivered by q0 ðtÞ and q0 ðtÞ. The development of computational schemes is not a difficult task considering that the problem is in one space dimension. The implementation can be based on classical collocation methods consisting in interpolating the dependent variable, in a suitable set of collocations point, by Lagrange polynomials or Sinc functions [10]. Subsequently the initial-boundary value problem is transferred into an initial value problem for ordinary differential equations, where boundary conditions are implemented in the first and last point of the collocation. The contents of [10] precisely refers to transport nonlinear diffusion equations analogous to that proposed in this paper, therefore it is a useful reference. Of course an appropriate collocation method needs to be selected by taking into account the space structure of the decomposition into the three domains proposed in Section 3. It is useful remarking that boundary conditions appear in the structure of the equation, and that the collocation method naturally implements them from the information received in the first and last node of the collocation. 4. Critical analysis and perspectives towards computing The contents of this paper are proposed in view of developments of computational schemes and subsequent comparisons with empirical data, which, as already mentioned, provide an information of the flow which is more accurate than that for the density. This aspect of the modeling approach is even more important in the presence of tollgates where measurement of the flow can be technically achieved, while that of the density does not capture the whole complexity of stop and go dynamics. Moreover this approach includes the natural trend of the driver-vehicle subsystem toward the flow near tollgates, which was not considered in previous approaches [17,4]. The contents focused on first order models. The author is well aware that this strategy has the disadvantage of involving a rather simple class of models, where some of the phenomena that can be depicted by higher order models can be lost. On the

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other hand, the flux limited approach included in the perceived density provides a realistic description of the diffusion phenomena otherwise not realistic due to the resulting infinite propagation speed. This reasoning identifies a natural research perspective consisting in the development of the approach to higher order models such as those reviewed in [2]. Possibly it can be extended to kinetic type models [12,5], still this perspective has to face additional difficulties related to the structure of kinetic equations that provide a description at the micro-scale, while measurements at the boundary are delivered at the macro-scale. Finally, let us stress that a natural development of the approach presented in this paper can be addressed to the modeling of crowd dynamics such as those reviewed in [3]. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2014.02.038. References [1] N. Bellomo, V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow, C.R. Mécanique 333 (2005) 843–851. [2] N. Bellomo, C. Dogbé, On the modelling of traffic and crowds: a survey of models, speculations, and perspectives, SIAM Rev. 53 (3) (2011) 409–463. [3] N. Bellomo, B. Piccoli, A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci. 22 (2) (2012) 1230004. [4] N. Bellomo, E. De Angelis, L. Graziano, A. Romano, Solution of nonlinear problems in applied sciences by generalized collocation methods and mathematica, Comput. Math. Appl. 41 (10–11) (2001) 1343–1364. [5] A. Bellouquid, E. De Angelis, L. Fermo, Towards the modeling of vehicular traffic as a complex system: a kinetic theory approach, Math. Models Methods Appl. Sci. 22 (1) (2012) 1140003. [6] G. Coclite, M. Garavello, B. Piccoli, Traffic flow on road networks, SIAM J. Math. Anal. 36 (2004) 1882–1886. [7] V. Coscia, On a closure of mass conservation equation and stability analysis in the mathematical theory of vehicular traffic flow, C.R. Mécanique 332 (8) (2004) 585–590. [8] C. D’Apice, B. Piccoli, Vertex flow models for network traffic, Math. Models Methods Appl. Sci. 18 (2008) 1299–1316. [9] E. De Angelis, Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems, Math. Comput. Model. 29 (1999) 83–95. [10] E. De Angelis, R. Revelli, L. Ridolfi, Transport–diffusion models with nonlinear boundary conditions and solution by generalized collocation methods, Comput. Math. Appl. 58 (3) (2009) 558–565. [11] M. Dolfin, Fron vehicle-driver behaviors to first order traffic flow macroscopic models, Appl. Math. Lett. 25 (2012) 2162–2167. [12] L. Fermo, A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math. 73 (2013) 1533–1556. [13] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73 (3) (2001) 1067–1141. [14] D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, Eur. Phys. J. B. 69 (2009) 539–548. [15] M. Herty, A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput. 25 (2003) 1066–1087. [16] B. Kerner, The Physics of Traffic, Empirical Freeway Pattern Features, Engineering Applications and Theory, Springer, Berlin, 2004. [17] A. Marasco, Nonlinear hydrodynamic models of traffic flow in the presence of tollgates, Math. Comput. Model. 35 (5–6) (1999) 549–559. [18] G.H. Peng, A new lattice model of traffic flow with the consideration of individual difference of anticipation driving behavior, Commun. Nonlinear Sci. Numer. Simul. 18 (10) (2013) 2801–2806. [19] G.H. Peng, A new lattice model of two-lane traffic flow with the consideration of optimal current difference, Commun. Nonlinear Sci. Numer. Simul. 18 (3) (2013) 559–566. [20] G.H. Peng, A new lattice model of the traffic flow with the consideration of the driver anticipation effect in a two-lane system, Nonlinear Dyn. 73 (1–2) (2013) 1035–1043. [21] G.H. Peng, W. Song, Y.J. Peng, S.H. Wang, A novel macro model of traffic flow with the consideration of anticipation optimal velocity, Physica A (2013). . [22] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag, 2007.