Numerical simulation of some macroscopic mathematical models of traffic flow. Comparative study

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1 Numerical simulation of some macroscopic mathematical models of traffic flow. Comparative study A. Lakhouili 1, El. Essoufi 1, H. Medromi 2, M. Mansouri 1 1 Hassan 1 st University, FST Settat, Morocco 2 Higher National School of Electricity and Mechanics. Casablanca. Morocco. ABSTRACT In this paper, some macroscopic mathematical models used urban traffic are presented and compared. The models are formulated in a continuous space-time framework before being discretized in space and time. Then we will do a theoretical study of the Riemann p r o b l e m in order to achieve a numerical s c h e m e says Godunov scheme which is the key to our modeling. Keywords: Traffic flow, Riemann problem, weak solution, Godunov scheme, entropic solution. 1. INTRODUCTION The modeling of urban road traffic is mainly done through three different approaches. The first approach is macroscopic in which the used models describe globally de flow traffic flow for the resolution of planning issues from a strategic point of view. The second approach called Microscopic having the advantage of treatment of interactions between individual vehicles and taking into account a large number of parameters associated with them and as a result it is more adapted to the reality of urban traffic. However, and given the large number of vehicles and related parameters, treatment becomes more difficult and calibration of used models very difficult. Recently, several research projects have focused on a new concept of hybrid modeling using jointly the two older models in order to improve the modeling by combining their strengths and overcoming their difficulties. In the first, we present the continuous approach which is the basis of the macroscopic modeling. In this approach to the modeling of vehicular traffic the flow of cars along a road is assimilated to the flow of fluid particles, for which suitable balance or conservation laws can be written. For this reason, macroscopic models are often called in the present context hydrodynamic models. All models include the conservation equation and a nonlinear, dynamic volumedensity relationship. In this paper, some macroscopic mathematical models are presented and compared. The models are formulated in a continuous space-time framework before being discretized in space and time. Then we will do a theoretical study of the Riemann p r o b l e m in order to achieve a numerical s c h e m e says Godunov scheme which is the key to our modeling. 2. TRAFFIC FLOW MODELS Macroscopic modeling is based on the idea, originally due in the fifties to Lighthill and Whitham, and, independently, to Richards, that the classical Euler and Navier-Stokes equations of fluid dynamics describing the flow of fluids could also describe the motion of cars along a road, provided a large-scale point of view is adopted so as to consider cars as small particles and their density as the main quantity to be looked at. This analogy remains nowadays in all macroscopic models of vehicular traffic, as terms like traffic pressure, traffic flow, traffic waves demonstrate. In the macroscopic approach to the modeling of vehicular traffic the flow of cars along a road is assimilated to the flow of fluid particles, for which suitable balance or conservation laws can be written. For this reason, macroscopic models are often called in the present context hydrodynamic models. The main dependent variables introduced to describe mathematically the problem are the density of cars ρ, their average velocity u at time t in the point x and the flux q given by The basic evolution equation translates the principle of conservation of the vehicles: t ρ + x q = 0 (1) Volume 3 Issue 11 November 2015 Page 1

2 ρ(0, x) = ρ 0 (x) (2) The equation (1) is a so-called conservation law since it expresses the conservation of the number of cars. The equation (2) expresses initial c on d i t i on. It can be questioned that Eq. (1) does not give rise by itself to a selfconsistent mathematical m odel, as it involves simultaneously two variables, ρ and u. We also need an equation for the velocity u. We assume that u only depends on ρ. If the highway is empty (ρ = 0), we will drive with maximal velocity u = u max ; in heavy traffic we will slow down and will stop (u = 0) in a tailback where the cars are bumper to bumper (ρ = ρ max ). In the next, we present som e traffic flow models. The first model has been already presented. The first and simplest model is the linear relation of Greenshields or the Lighthill-Whitham-Richards model: The prototype of all fluxes complying with the above assumptions is the parabolic profile firstly proposed by Lighthill and Whitham [10] and then, independently, by Richards [13], which gives rise to the so-called LWR model: A generalization of equation (3) and (4) is provided by the Greenshield model: Another example of fundamental diagram satisfying previous assumptions but which generates an unbounded velocity diagram for ρ 0+ is due to Greenberg: In this model it is assumed that the velocity of the vehicles can be very large for low densities: Moreover, other closure relations of the mass conservation equation for first order models are discussed in Bellomo and Coscia [2]. Here we simply mention a further diagram due to Bonzani and Mussone [3] For all previous models, a relation of the form q = f(ρ) is named in this context fundamental diagram. Note that if > 0 denotes the maximum vehicle density allowed along the road according, for instance, to the capacity, i.e., the maximum sustainable occupancy, of the latter, the function f is often required: (1) To be monotonically increasing from ρ = 0 up to a certain density value s (0, ); (2) To be decreasing for ; (3) To have as unique maximum point in [0, ]; (4) To be concave in the interval [0, ]; 3. NUMERICAL SCHEME The traffic flow models are nonlinear, s o we want to study what can happen when we discretize nonlinear i equations. Hence, we derive a conservative and consistent scheme which avoids the above problem, the so-called Gudonov scheme. We discretize the (x; t)-plane by the mesh (xi ; tn ) with xi = ih, i Z and tn = nk, n N. For simplicity of presentation we take a uniform mesh with h and k constant, provided the interaction is entirely contained within a mesh cell. T he discussed methods can be easily extended to non-uniform m e s h e s. We are looking for approximations t o t h e s o l u t i o n Volume 3 Issue 11 November 2015 Page 2

3 at the discrete grid points. The idea is as follows. Let q be a convex C 2 function. The idea of the method is to approximate t h e solution ρ(x; tn ) of the conservation law (2) by a piecewise constant function ρ n (x; tn ) and to determine the approximate solution ρ n (x; t) by solving the Riemann problem in the interval t [tn, tn+1 ] is exact solution at time t n+1. After obtaining this solution, we define the approximate solution at time tn+1 by averaging this exact solution at time tn+1 : (8) where data by: These values are then used to define the new piecewise constant Where h= and k= And the process repeats. In fact, we can allow the waves to interact d u r i n g the time step, provided the interaction is entirely contained within a mesh cell. This leads to the condition C F L (figure 1): In practice, t h i s algorithm is considerably simplified since the above integral can be computed explicitly. 4.Numerical result and discussion Greenshields model The solution is well modeled whatever its type: shock wave or wave relaxation (figure 2). As to time, we get closer Solutions the exact solution. From successive compilations accompanied by changes in the value of the calculation time T that depends on the fixed time t in the program (it is the stopping criterion of the construction of the approximate solution), it is found that the calculation time T in the case of shock reaches its maximum at T = 13 seconds, and in the case of relaxation achieved its maximum at T = 2.82 seconds. Volume 3 Issue 11 November 2015 Page 3

4 Figure 2: Shock and relaxation wave in Greenshields model Drew Model Regarding accuracy, Drew model is similar to the model Greenshields because it also gives closer to the exact solution solutions but regarding the speed, it is noted that the computing time T in the case shock peaks at T = 22,87 seconds, and in the case of relaxation there reaches its peak at T = 19,68 seconds. This means that the model Drew is slower than the model of Greenshields. Figure 3: Shock and relaxation wave in Drew model Edie model This model contains only a fluid region, and the critical density p is equal to the Maximum density pm. Despite the absence of the congested area we can test both studied waves. The Edie model differs from the other two models: Greenshields and Drew, because it is not entropic in the case of shock: it was not always good physical solution. The fact that, whatever the considered stopper, the latter recedes always. For this, the model remains somewhat limited for the shock wave solution. However, for the relaxation wave, the model is more efficient and the computing time T reaches its maximum at 18 seconds. Figure 4: Shock and relaxation wave in Edie model Volume 3 Issue 11 November 2015 Page 4

5 Chandler model This model, contrary to that of Edie, contains only a congested area and the critical density is zero. In addition, the inverse function of the derivative of flow q 'does not exist since this model is linear. Therefore, it can only test the shock waves. We remark that the computing time T reaches its maximum at 2,76 seconds. Therefore, the model speed is excellent, while the accuracy is influenced by the choice of q m Drake model Figure 5: Shock and relaxation wave in Chandler model To test this model should be chosen a priori a value that describes the critical density, eg = 2,3. Captured Figures show that the approximation is better for this model with good accuracy. Concerning speed, the computing time T in the case shock peaks at T = 12,11 seconds, which is similar to the Drew model, and in the case of relaxing it reaches its maximum at T = 20 seconds, and in this case this model resembles that of Greenshields. Figure 6: Shock and relaxation wave in Drake model 5.Conclusion We conducted a homogeneous macroscopic traffic modeling based on the models of the first order. In parallel with the theoretical study we did, we compared the first order models using Godunov diagram and the initial conditions of Riemann type. However, the Godunov scheme is well adapted with this kind of modeling, in condition to choose the right step mesh h. It turns out that the approximation is better with approached solutions that have precision and coherence with real situations observed on the road, either in a collision or relaxation situation. The Chandler model will be used to control maximum flow q m in the collision case, it is very fast, but its accuracy is influenced by the choice of q m. The Edie model is limited by problems in the case of the shock wave, this model is the most suitable in the relaxing case when there is no exigency of the accuracy of results and when a rapid calculation is whished. Drew's model is similar to Greenshields on precision, but we prefer to use the Greenshields when we are seeking optimum calculation time and precision. Thus, the two models Greenshields and Drew, are adapted to the shock position. Therefore, they will be used in both cases shock and detents. Volume 3 Issue 11 November 2015 Page 5

6 Références [1]. A. Aw and M. Rascle: Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60 (2000), , Vol. 355, Issue 2, 15 July 2009, Pages [2]. N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow, C. R. Mecanique 333 (2005), [3]. I. Bonzani and L. Mussone, Stochastic modelling of traffic flow, Math. Comput. Modelling 36 (2002), no. 1-2, [4]. C. Daganzo: Requiem for second-order approximation t o traffic flow. Transport.Res. B 29 (1995), [5]. M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations 31 (2006), no. 1-3, [6]. M. Garavello and B. Piccoli : Traffic flow on networks, AIMS Series on Applied Mathematics, vol. 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models. [7]. B. S. Kerner and S. L. Klenov, A microscopic model for phase transitions in traffic flow, J. Phys. A 35 (2002), no. 3, L31 L43. [8]. B. S. Kerner, Phase transitions in traffic flow, Traffic and Granular Flow 99 (D. Helbing, H. Hermann, M. Schreckenberg, and D. E. Wolf, eds.), Springer-Verlag, New York, 2000, pp [9]. R. Leveque: Numerical Methods for Conservation L a ws. Birkhuser, Basel, [10]. LIGHTHILL. M.J, WHITHAM. G.B., On kinematic waves II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society A, vol. 229, pp [11]. B. Piccoliand A. Tosin : a review of continuum mathematical model of vehicular traffic, [12]. The physics of traffic, Springer, Berlin, [13]. RICHARDS. P.I, Shockwaves on the highway, Operations research, vol. 4, pp Volume 3 Issue 11 November 2015 Page 6