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This paper concerns the traveling wave formation in macroscopic traffic flow models. The dynamics involved in this problem is described following a close analogy to compressible fluid dynamics. It is well known that vehicle clusters appear along a highway when the homogenous steady state taken as a reference is linearly unstable. The cluster properties are determined in an approximate way in terms of the parameters proper to each model and are compared between them.

The study of density waves in traffic flow has constituted a subject of interest mainly in relation to the cluster formation. Clusters appear in real traffic only under certain conditions, then it is interesting to understand the mechanisms inducing such a phenomenon. Microscopic models such as the car-following models have studied numerically the structure as well as the conditions under which traveling waves can be formed, see for example [1-3]. On the other hand, we know that the microscopic approach is not the only one to study traffic flow phenomena. The macroscopic traffic flow models represent a possible approach when applied to study vehicle behavior in a highway. The development of such models can be done either in a phenomenological way [4-13] or taking as a starting point a kinetic equation [14-17] from which the macroscopic equations can be obtained. Besides, the car-following class of models can be written as continuum equations [

The macroscopic traffic flow models, called as GM models, consider the motion of vehicles in highways as the compressible flow of a fluid. This means that the number of vehicles is large enough to consider the system as a continuum, in contrast with microscopic models where each vehicle is followed in its motion. The continuum approach describes the dynamics in the system by means of averaged variables such as the density, the speed, the speed variance and, some others. The number of macroscopic variables needed to understand the behavior varies according to the level of description we want to give. It should be noted that we consider all vehicles traveling in one direction, along a highway with one lane, this fact makes the problem one-dimensional. Also, we consider the motion in a closed circuit in order to have periodic boundary conditions. In this work we will consider three models [8,13,18], which share the structure of the balance equations for the relevant variables. First of all, we have the continuity equation for the density

valid for the three models. The average speed satisfies a balance equation, in the case of models coming from a direct analogy with Navier-Stokes compressible fluid mechanics, such as the Helbing’s improved model (H) and the Kerner-Konhäuser model (K), it means

where is the traffic pressure and the average relaxation time. In the case of the model called as Continuum Car-Following (CCF) it comes from a discrete traffic model, in such a case the speed equation shares the structure, but it cannot be written as in Equation (2) as emphasized in Section 4. Model H also considers the speed variance dynamics as relevant in the description, it has the structure of a usual balance equation and will be given afterwards. All these models have the contribution coming from the fundamental diagram, which represents the speed profile to which the drivers tend with a relaxation time given by. The fundamental diagram gives us a relation between the speed and the density when the system is in a homogeneous steady state. It considers all values between zero and their maximums in the speed and the density, meaning that such a relation exists even in the congested traffic region. It seems that this hypothesis is responsible for the lack of transition from free to synchronized phase, in GM models. The fundamental diagram, as well as the relaxation time are taken from the empirical observations. The models studied here will take the speed given by the fundamental diagram which tends to the maximum speed allowed in the highway at low densities and, tends to zero when the density is near the maximum density determined by the bumper-tobumper distance. In this work we will take the usual correlation for the fundamental diagram [8,9] which is given as

where , , with a maximum speed and the maximum density. The boundary conditions to simulate the solution are taken as periodic

they correspond to a closed circuit with a length given as. On the other hand, the initial conditions are chosen as follows

where the constants will be given for each illustrated case. The homogenous steady state determined by is a solution of the equations of motion in model H and, in the case of model K and CCF the corresponding state is given through The linear stability conditions associated to this state give us a way to begin the study of the mathematical properties contained in them. The linear stability conditions in these models are studied by means of a linearization of the equations of motion around the homogeneous steady state (called as the equilibrium state), then

where and do not depend on position and time, is the wave vector and can be a complex function of the wave vector. The linear stability around the homogeneous steady state is granted when.The stability conditions determined in this way will be different for each model and they will be discussed in each particular case.

When the density is chosen in the stable region, the perturbations around it decrease and the vehicles recover the homogenous steady state. On the other hand, if we are working in the unstable region, the perturbations grow and it is necessary to go into a deeper analysis.

To begin our treatment, we consider the macroscopic model introduced by Helbing in 1995 [

where the traffic pressure is proposed with a viscosity coefficient and, a size correction for vehicles where is the vehicle length and is the preferred headway of the driver, in such a way that represents a safe distance. These assumptions drive to a traffic pressure given as

Notice that the traffic pressure resembles the corresponding Navier-Newton constitutive equation, where we have a term playing the role of the hydrostatic pressure, a viscosity coefficient with a size correction and, the speed gradient. On the other hand, the quantity in Equation (10) represents the skewness in the speed distribution and can be seen as the analogous of the heat flow in compressible flow, it contains a kind of thermal conductivity coefficient and the size correction, they are given as follows

the quantities and are constants. It is worth noticing that the structure of the traffic pressure and, are very similar to the fluxes in usual fluid mechanics, however in traffic flow they do not have the same meaning. The finite size correction when and, it has the effect to enhance the kinetic coefficients with a quantity which resembles the situation in a dense fluid. In contrast with Helbing’s work, in this paper the equilibrium variance is given through the variance prefactor and the fundamental diagram,

the dimensionless variance prefactor is taken from correlations based on the empirical observations [

where and are constants.

Now, the homogeneous steady state is determined by which are constants and satisfy the set of the equations of motion, as mentioned in Section 2. The linear stability analysis goes along the steps given in Equations (7)-(9). Also, the real parts in will determine the stability conditions, in fact, when we have stability. The real parts are developed in terms of the wave vector and the leading terms are given as follows,

where we have written

and to shorten the notation. The quantities are both related with the finite size of vehicles. On the other hand, the imaginary parts determine the speed of long wave length perturbations, they are given as

. Here the speed is given as

and it is determined by the characteristics in the fundamental diagram. When we take the expression given in Equation (3), we can see that can be positive or negative as a function of The negative value of the imaginary parts for the three roots in Equation (15) is shown in

As a third comment, we notice that the real parts in roots define a time scale proportional to, which does not tend to zero in the limit when. As a summary, the real parts in the dispersion relation roots, define two time scales in the problem, one of them determines the stability and the other one is given according to the model. When the stability condition is satisfied as an equality we obtain marginal stability.

In this model, the marginal stability reduces to a point corresponding to an equilibrium density

. It means that for densities

the homogeneous steady state is not linearly stable. In this point, it is important to emphasize that in model H, the homogeneous congested traffic state (HCT) is not present. This characteristic makes model H different from other GM models, where the stability region is bounded by a coexistence curve allowing a HCT state at high densities.

The simulation is done according to the expressions given in Section 2, with the following data

and the speed variance calculated according to Equations (13)-(14) is given by

.

It is important to notice that the density chosen to implement the simulation corresponds to the linearly unstable region. As a result we have obtained a density profile in which the maximum density is less than

and the speed is always positive. This result means that once the perturbation has occurred it grows and then, after a transient time interval, a permanent density profile is obtained. Also, the presence of this profile indicates a cluster (J) formation propagating along the highway. The cluster speed measured shows that it remains constant, once the traveling wave has acquired its permanent profile. The data given in the simulation imply the existence of a cluster traveling upstream with a constant speed. Lastly, the characteristics of the cluster, such as the amplitude and width depend on the values taken for the model parameters, as will be shown in Section 6.

As a second example, in

a bigger value for.

Let us consider two macroscopic models which share the continuity and speed equations of motion, but do not consider the speed variance as determined by the dynamics.

In this model the traffic pressure is given according to a close analogy with fluid mechanics, in fact it corresponds to a particular case in model H. It is obtained when we neglect the size of vehicles, take a constant value for the speed variance, in the traffic pressure and a viscosity. Then the traffic pressure is given as

and it resembles to the Navier-Newton constitutive equation for a viscous fluid. In our case the coefficient is the analogous of the bulk viscosity, due to the compressible character in the flow. The first term is the analogous of the hydrostatic pressure and, in this case is represented by the term.The direct substitution of the traffic pressure in Equation (2) gives us

where we can see in a clear way the analogy with the Navier-Stokes equation. Notice that the speed variance is not taken as a relevant variable, then we only need two equations of motion to describe the system.

According to the stability considerations given in Section 2, we linearize the set of Equations (1), (19) around the equilibrium state and calculate the real parts of roots, hence

The imaginary parts of roots to lowest order in the wave vector are given as, where gives us the propagation of long wave length perturbation waves, its behavior can be seen in

The results shown in Equations (20), (21) show us immediately that in this model there exist two time scales, one is determined by the relaxation time and the other one depends on the fundamental diagram and the value for the variance. Besides they depend on the wave vector. In fact, Equation (20) determines the marginal stability and it contains the reference density, in such a way that we can consider a kind of coexistence curve which separates the space determined by in stability regions.

the stability region corresponds to

and it is located outside the curve. This figure shows in a clear way that model K allows the existence of HCT states. It is important to notice that the roots are given to leading orders in the wave vector.

As a second step in the K model analysis, we recall that the stability regions are determined within the frame on the linearization of the equations of motion. It means that beyond the linear regime, this analysis cannot be taken. In fact, the present instabilities do not show themselves when we simulate the complete model, even when we work around densities chosen in the unstable region.

The numerical simulation of this model starts with the initial conditions shown in Equations (5), (6) with ,

and the equilibrium density in the unstable region when. Also, we take periodic boundary conditions for the density and the speed, as given in Equation (4).

In all cases the speed and density profiles are coupled closely. The maximum (minimum) in density corresponds to a minimum (maximum) in the speed.

There are several microscopic models to study traffic flow, one of the most common class corresponds to the

car-following structure. Though these models follow each vehicle while they are moving, Berg, et al. [

where we have written the relaxation time instead of the sensibility factor as it is usual in this model. Equation (22) can also be written in a similar way as the Navier-Stokes equation,

In contrast with the model K, now we have density dependence in the term equivalent to the hydrostatic pressure while the viscosity term depends on the density gradient instead of the speed gradient. In spite of these differences, the structure of the equation contains the same characteristics as Equation (19). The linear stability analysis can be done with the same procedure to obtain

where it is shown the two time scales and quantity is given in Equation (16). Consequently we also have a coexistence curve separating the space according to the stability characteristics.

We remark that models K and CCF have a coexistence curve which bounds the instability region. Stability occurs outside the marginal stability curve shown in Figures 5, 7 giving place to the HCT states at the high density region.

A way to go further in the search for an explanation to this behavior, is provided by the method proposed by Berg and Woods [

To begin the procedure, the equations of motion (1, 2, 22, 10) are written in a reference frame moving with constant speed. We introduce the variable and, all derivatives will be indicated with a sub index. The continuity equation is then written as

which can be integrated, the speed Equation (2) is written in terms of the flux

where in agreement with the continuity Equation (26). We recall that Helbing and Kerner-Konhäuser’s models share the structure of the speed equation, the difference between them is given through the traffic pressure expressions. In a similar way, the speed equation in the continuum car-following method Equation (23) is also written in the new reference frame,

lastly, the speed variance equation is written as

The first step in the iterative process is done with the consideration of a solution up to order zero in the -derivatives. It is clear that this solution is given as

To continue in a clear way, let us consider first the Helbing’s model where the -derivative for the traffic pressure is given as

The substitution of Equation (31) in the left hand side of Equation (27) gives us a value for the first order iteration in the flux,

where the bilinear or quadratic terms like have been neglected to this order. Now, the speed given according to the fundamental diagram is developed around the equilibrium density up to second order terms,

and we have called the density deviation as.

The direct substitution of Equation (32) and the expansion in Equation (26) drives us to,

where is given in Equation (16) and

is determined in terms of the fundamental diagram and the speed. It should be noted that Equation (33) does not represent the complete solution as described in the simulation, instead it is an approximation of the model in which some small terms have been neglected. For example, the fundamental diagram contains terms of all order in, there are derivatives of higher order than the ones considered in this approximation, also, some nonlinear terms in the derivatives are neglected. Equation (33) is the well known Korteweg-de Vries (KdV) equation for the density when written in the moving reference system [

where we recall that the quantities are evaluated in the density, besides. This solution is obtained by means of direct integration and two integration constants are taken as zero. The quantity corresponds to an integration constant and it is interpreted as the soliton center. Obviously, the soliton amplitude is given by and it must be positive. Also, the quotient must be positive, when both conditions are valid simultaneously, they guarantee that the soliton amplitude be positive and the width a real quantity, as we will see soon.

We now apply the same method to model K, taking the speed Equation (28) we find the first order approximation in the flux,

Also, we have neglected second order terms in the -derivatives. Now, we follow the same steps as in model H and find the corresponding KdV equation in this case,

which is essentially the same equation as in model H, the only difference being the presence of the vehicle size and safe distance as represented by.

Now, let us consider the set of Equations (26), (27) in the case corresponding to the CCF model. The iterative procedure follows the same steps to obtain the first order correction to the flux, the result is given as,

The procedure described above is applied to obtain the evolution of the density in the moving reference frame, so

where is given before. Equation (39) represents a soliton with the same structure as given in Equation (33) with some differences to be compared later.

Lastly, the variance corresponding to model H in this approximation can be written as

and we notice that it is not coupled with other quantities but the deviation in the density.

To compare the results obtained with the three models, let us define the soliton width, with respect to its center, and in terms of the density profile along the highway length,

This quantity can be calculated for the three models and the results are shown in the following table.

All of them look very similar however, they contain explicitly the parameters introduced by the model itself, the speed corresponding to the traveling wave and, the characteristics in the fundamental diagram. Notice that model H contains the size of vehicles and the safe distance through the quantity.

width (km)

The regions where the soliton solutions exist change according to the value of the equilibrium density and the soliton speed, which is the speed of the reference frame. In fact, the simulation results show the meaning that the cluster travels upstream. In

In a similar way we can construct the line allowing the soliton existence in the CCF model, where we see that there is only one line separating regions and, the soliton can travel upstream or downstream.

The macroscopic modeling approach to traffic flow has shown to reproduce some characteristics of the behavior of vehicles in a highway. In particular, the model H does not present the HCT state in contrast to models K and, CCF. The three models take the same fundamental diagram and in the linearly unstable region allow the formation of clusters which can be seen as wide moving jams (J). Besides the properties shown in the numerical simulation, the iterative procedure has allowed us to find, in an analytical way, the approximate structure of such clusters. Numerical calculations have been done to show this behavior in microscopic models, however, it is interesting to show that models including the size of vehicle behave in the same way. Certainly, the calculation is an approximate one, but it gives the qualitative properties of the clusters. The comparison between these models

has shown that the macroscopic analogy with compressible fluid mechanics is fruitful to understand some characteristics in traffic.