No. 11 Analysis of the stability and density waves for trafc flow 119 where the function f sti represents the response to the stimulus received by the

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1 Vol 11 No 11, November 00 cfl 00 Chin. Phys. Soc /00/11(11)/ Chinese Physics and IOP Publishing Ltd Analysis of the stability and density waves for trafc flow * Xue Yu( ) Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 0007, China Department of Physics, Guangxi University, Nanning 5000, China (eceived 9 March 00; revised manuscript received 18 June 00) In this paper, the optimal velocity model of trafc is extended to take into account the relative velocity. The stability and density waves for trafc flow are investigated analytically with the perturbation method. The stability criterion is derived by the linear stability analysis. It is shown that the triangular shock wave, soliton wave and kink wave appear respectively in our model for density waves in the three regions: stable, metastable and unstable regions. These correspond to the solutions of the Burgers equation, Kortewegde Vries equation and modied Kortewegde Vries equation. The analytical results are conrmed to be in good agreement with those of numerical simulation. All the results indicate that the interaction of a car with relative velocity can affect the stability of the trafc flow and raise critical density. Keywords: car-following model, trafc flow, density wave, relative velocity PACC: 0550, 050, Introduction Trafc problems have attracted considerable attention during the past decades. Various trafc models have been developed and numerous empirical observations have been reported. [1] A car-following theory called the optimal velocity (OV) model was proposed by Bando [] et al in 1995, which can describe the trafc-free flow and jam, and can be adapted to use the perturbation methods to analyse the trafc density wave. [5] Kerner and Konhaüser [] have found the single-pulse density wave in the numerical simulation with the hydrodynamic trafc model. Kurtze and Hong [4] have proven that the single-pulse density wave is a soliton. Komatsu and Sasa [5] derived the modied Kortewegde Vries (mkdv) equation from the OV model to describe trafc jams in terms of a kink density wave. Muramatsu and Nagatani [6] have shown that the solitonary density wave appears only near the neutral stability line. Nagatani [7] has concluded that the triangular shock wave, soliton wave and kink wave appear, respectively, for density waves in the three regions, stable, metastable and unstable regions, which are described by the solutions of the Burgers equation, KdV equation and mkdv equation. Based on the OV model, we have presented a simple optimal velocity car-following model with relative velocity in order to investigate the effects of relative velocity. [9;10] We found that the process from trafcfree flow to jam (free!jam) differs from that of trafc jam to dissolution (jam!free) in which there exists a hysteresis effect related to the metastable state. [11] Also, we have derived the mkdv equation to describe the trafc jam in the unstable region and to obtain the phase diagram (a; x) for various values of the response factor to the relative velocity, in which the stable, metastable and unstable regions are revealed. [9] In this paper, we propose a difference equation corresponding to the dynamic equation of the simple optimal velocity car-following model with the consideration of relative velocity. We analyse the stability and density wave for a trafc flow by means of the perturbation methods in the three regions, stable, metastable and unstable regions, and we derive the stability criterion. Finally, we compare our analytical results with those of numerical simulation..model and difference equation In general, the dynamic equation of the carfollowing model can be written as [1;8] ẍ n = f sti (v n ; x n ; v n ); (1) Λ Project supported by the National Natural Science Foundation of China (Grant No 19900).

2 No. 11 Analysis of the stability and density waves for trafc flow 119 where the function f sti represents the response to the stimulus received by the nth vehicle. Equation (1) gives the acceleration or deceleration of the nth vehicle, which is only determined by the surrounding trafc conditions. The stimulus may be composed of the velocity v n of the vehicle, the velocity difference (i.e. the relative velocity v n = v n+1 v n ), and the headway x n = x n+1 x n between successive vehicles. Combining with the optimal velocity model, we consider that the optimal velocity may be determined not only by the headway but also by the response to the stimulus of the relative velocity. Thus, we propose the following dynamic equation to describe the car-following process dx n (t + ) = V ( x n (t); v n (t)); () dt where x n (t) is the position of the nth vehicle, and V ( x n (t); v n (t)) is the optimal velocity function including variables of the headway and relative velocity. The idea is that a driver adjusts the vehicle velocity at time t according to the observed headway, and the relative velocity at time t to reach the optimal velocity, where is the time delay. Expanding both sides of Eq.() in Taylor series, we obtain d x n dt = a V ( x n (t)) dx n dt + v n (t); () where we assume that the optimal velocity is V ( x n (t), v n (t)) = V ( x n (t)) + v n (t); a is the sensitivity of a driver to the observed headway x n, a = 1 ; is the sensitivity of response to the stimulus v n and is taken to be =. It is assumed that is a constant independent of time, velocity and position, which is called the response factor to the relative velocity (0 1). By using the asymmetric forward difference, we rewrite Eq.() as x n (t +) =x n (t + )+V ( x n ) + [ x n (t + ) x n (t)]: (4) We choose the same optimal velocity function as that used by Bando et al [] V ( x n ) = v max ftanh( x n h c ) + tanh(h c )g; (5) where h c is the safety distance. The optimal velocity function has the following properties. It is a monotonically increasing function of the headway and has an upper bound (i.e. the maximal velocity). As x n! 1, the optimal velocity reaches its maximum V (1) = v max f1 + tanh(h c )g=. Furthermore, for h c fl 0;V(1) ο = v max and thus the maximal value of the optimal velocity is v max for x n fl h c fl 0. The optimal velocity function given by Eq.(5) has a turning point at x n = h c : V 00 (h c ) = d V ( x n ) d x j xn=hc = 0: n. Linear stability theory We apply the linear stability theory to analyse the trafc model described by Eq.(4). The procedure is similar to that used by Bando et al. [] We rst consider the stability of a uniform trafc flow. The uniform trafc flow is dened by such a state that all cars move with the optimal velocity V (h) and the identical headway h; the relative velocity v n is zero. The solution x n;0 (t) representing the uniform steady state is given by x n;0 (t) = hn + V (h)t; h = L N ; (6) where N is the total number of cars and h is the headway. Letting y n (t) be a small deviation from the steady state x n;0 (t), we have x n (t) = x n;0 (t)+y n (t): (7) Then the linearized equation is obtained y n (t +) =y n (t + )+V 0 (h) y n + [ y n (t + ) y n (t)]; (8) where V 0 (h) is the derivative of the optimal velocity function at x = h. By expanding y n ß exp(ikn+zt), the following equation of z is obtained (e z 1)(e z e ik + ) = V 0 (h)(e ik 1): (9) By expanding z = z 1 (ik)+z (ik) +, we have z 1 = V 0 (h); z = V 0 (h) [1 V 0 (h) +]: (10) If z is negative, the uniform steady state will become unstable, and the neutral stability criterion is given by V 0 (h) 1+ : (11a) For small disturbances, the uniform trafc flow will be unstable if V 0 (h) > 1+; (11b) a

3 110 Xue Yu Vol. 11 and the critical time delay is c = 1+. As the time delay < c, the driver can adjust his vehicle velocity to reach the optimal velocity. Otherwise, the trafc flow will be unstable, a trafc jam will occur, and the vehicle will be too slow to reach the optimal velocity. The stable criterion parameters (a; x; ) determine the stable state. Comparing with the results of Bando et al, [] we can nd that the relative velocity has improved the stability effect for trafc flow. [9;10] 4. Nonlinear analysis and density wave We consider respectively the slowly varying behaviours for long waves in the stable and unstable regions. We dene a small positive scaling parameter " near the neutral stability line as follows V 0 (1 + a s) fa " = as near the critical point = a as a 1 = " ; (1) 1 1 a ; = a ; (1) a = a c ; " = ac 1 1 a : (14) We wish to extract slow scales for the space variable n and the time variable t. For 0 < " 1, we dene the slow variables X and T X = " P (n + bt); T = " q t; (15) where b is a constant to be determined. We set the headway as x n = h + " m (X; T): (16) The three groups of values p=1, q=, m=1; p=1, q=, m= and p=1, q=, m=1 correspond respectively to the headways in the stable region, [7] near the neutral stability line [5] and in the unstable region. [4] Equation (4) can be rewritten as partial differential equation " p+m X + " T + " p+m X! q+m + T + "p+q+m X + " p+m 7 X + T + " p+q+m X! + "p+q+m T! + " 4p+m 15b T + T 4 4! 4! + " 5p+m 1b X + T 5 5! 5! = " p+m V 0 (h)@ X + " p+m V 0 X! + " p+m V 0 X + V 0 (h) X 4 4! + " p+m V 00 (h)! + " p+m V 00 (h) 4! + " p+m V 000 (h) X + " p+m V 00 X X + " p+m V 000 X + " p+m V 000 (h) X + " p+m X + X p+m (1 + b)b + X + X@ T + " 4p+m (1 + b)4 1 (b) X 4 4! + " 5p+m (1 + b)5 1 (b) X ; 5 5! (18) @ @ X@ V 0 = dv ( x) d x j x=h and V 000 = d V ( x)=d x j x=h. (1) For p=1, q=, m=1, we obtain the following nonlinear partial differential equation from Eq.(18) ρ V " [b V 0 (h)]@ X + 0 T + (1 + ff )V X V X = 0; (19a) by taking b = V 0. The second-order term of " is eliminated from Eq.(19a) and we have x n (t +) =[V ( x n+1 V ( x n )] + [ x n+1 (t + ) x n+1 (t) x n (t + )+ x n (t)]: T V X 1+ = V 0 (h) V X : (19b) By expanding Eq.(17) to the fth order of " and using Eqs.(15) and (16), we obtain the following nonlinear As x > h, we have V 00 (h) < 0. The coefcient 1+ V 0 (h) > 0 in the stable region satises

4 No. 11 Analysis of the stability and density waves for trafc flow 111 the stability criterion (11a). Thus, in the stable region, Eq.(19b) is just the Burgers equation. The solution of the Burgers equation is given by (X; T) = 1 jv 00 j X 1 ( n + n+1 ) 1 jv 00 jt ( n+1 n ) c 1 tanh 4jV 00 jt ( n+1 n )(X ο n ) ; (0) where c 1 = V 0 = V 0 = and ο n 's (n=1,,...,n) are the coordinates of the shock fronts, and n are the coordinates of the intersections of the slopes with the x-axis (n=1,,...,n). The propagation velocity of the triangular shock wave is given by v p = V 0 (h): (1) () For p=1, q=, m=, we obtain the following nonlinear partial differential equation from Eq.(18): " [b V 0 (h)]@ z + " T + X V 0 X =0: () By taking b = V 0, the third-order term of " is eliminated from Eq.(), and we have T = V 0 (h) V 0 (h)@ X : () Equation() is the diffusion equation without V 00 (h). Its solution in the stable region V 0 (h) < 1+ is given by 1 (X; T) = s 1+ 4ß V 0 (h) exp 1+ 4 X V 0 (h) T : (4) Equations (0) and (4) indicate that the density wave of the trafc flow is described by the Burgers equation or the diffusion equation in the stable region. As t! 1, (X; T)! 0, the trafc flow will be in a homogeneous distribution. As V 0 (h) 1+, the diffusion in the unstable state leads to a trafc jam. () For p=1, q=, m=1, we obtain the following nonlinear partial differential equation from Eq.(18) " [b V 0 (h)]@ X + " X V 0 X V 00 X X + " T + 7b (1 + X V 0 X V 00 X 4 V 000 X + O(" 5 ) = 0; (5) near the critical point h = h c, V 00 (h c ) = 0. By taking b = V 0 (h c ), = c = 1 + ", and c = 1+ the V 0 second-order term of " is eliminated from Eq.(5). Using Eq.(14), we have 1+4 T V X 7 V 000 X + O(") = 0: (6) In order to derive the regularized equation, we make the following transformations T = V 0 T; 7 = ( = 5 ) V 0 0 ; (7) 9V 000 where and 0 1, and we obtain the regularized 0 T X 0 X 0 + O(") = 0; (8) if we ignore O(") in Eq.(8). This is just the mkdv equation with a kink solution as the desired solution (X; T) = ( ) V 0 c 9V r 000 c tanh X 1= V 0 ct ; 7 (9) 7(1 + ff) where c =. Thus, we obtain the kink solution of the headway x n (t) = h c ± r ac a 1 (X; T): (0) The mkdv equation is derived near the critical point (h < h c ), and the kink wave appears as the density wave in the unstable region, which is described by the solution of the mkdv equation.

5 11 Xue Yu Vol. 11 (4) For p=1, q=, m=, we obtain the following nonlinear partial r differential equation near the neutral as stability " = a 1 1;a s = V 0 (h) from Eq.(18) 1 + " [b V 0 (h)]@ X b + " 4! + " X V 0 X! T + 7b V 00 X! + " X V 0 X (1 + X + 15b4 X 4 4! V 00 X X (1 + b)4 1 (b) X 4 4! = 0: (1) 5. Numerical simulation We have analysed the stability and density wave of a trafc flow by means of perturbation methods in the different regions. It is shown that the triangular shock wave, soliton wave and kink wave respectively appear in our model as the density waves in the three regions stable, metastability and unstable regions which are respectively described by the solution of the Burgers equation, KdV equation and mkdv equation. We further conrm whether there exist three different trafc density waves in these regions by numerical simulation. It is assumed that the system has N vehicles distributed on the road under a periodic boundary condition. We simulate the density wave of the trafc flow according to Eqs.(4) and (17). Firstly, we simulate the triangular shock wave in the stable region. The initial headway is chosen as follows x n = 7:0; x n (0) = x n (1) = x 0 :0 By taking b = V 0 (h), the third-order term of " is eliminated from Eq.(1). Inserting Eqs.(1) and (1) into Eq.(1), we obtain the KdV equation with the perturbed term. If we ignore O(") in Eq.(1), it is just the KdV equation with a soliton solution as the desired solution (5 + 1)(1 T V X 7 V 00 (h)@ X = 0: () We perform the following transformations (1 n N=); N = 00; x n (0) = x n (1) = x 0 +:0 (N= < n N); where x 0 is the average headway. Figure 1 indicates that the headway prole at t = 10 4 is the triangular shock wave in the stable region. Spacetime evolution of the headway is shown in Fig. after t = There are two points c and d, which represent respectively the shock front and the intersections of the slopes. r (5 +1)(1 ) T = V 7 0 jgj T 0 ; r (5 + 1)(1 ) X = V 7 0 jgj X 0 ; = 1 V 00 jgj0 ; where g is a negative constant. We T 0 0 X + X 0 = 0: () Thus, we obtain the soliton solution of the headway 8V 0 x n (t) =h + 7(5 +1)(1 )V 00 a s a 1 v ( u t 7 a s sech a 1 14 a s h + 1+ a 1 V t ) 0 : (4) 9 Fig.1. Headway prole of the triangular shock wave at t=10 4. Points c and d represent, respectively, the shock front and the intersections of the slope. Fig.. Spacetime evolution of the headway after t = 10 4 when the initial density prole has the kink anti-kink form in the freely moving phase.

6 No. 11 Analysis of the stability and density waves for trafc flow 11 We also verify that the initial kinkanti-kink prole of the headway decays from the triangular shock wave to the uniform flow. Next, the simulation is carried out to validate the existence of the soliton wave described by the KdV equation in the metastable region. The initial headway is chosen as x n (0) = x n (1) = 7:0; N N n 6= ; +1 : N = 00; to the relative velocity. The initial headway is N N x n (0) = x n (1) = 4:0; n 6= ; +1 : N = 00; h c = 4:0 x n (0) = x n (1) = 4:0 = 4:0 0:1 n N ; N x n (0) = x n (1) = 4:0 = 4:0+0:1 n = +1 : x n (0) = x n (1) = 7:0 0: n N ; N x n (0) = x n (1) = 7:0+0: n = +1 : The pattern in Fig. shows clearly the soliton wave. Finally, we study the kink density wave appearing as a trafc jam and discuss the spacetime evolution of the headway for various values of the response factor j Fig.. Spacetime evolution of the headway after t = 10 4 when the initial prole has a small perturbation (a = :0; h c = 4:0; x 0 = 7:0). Fig.4. Spacetime evolution of the headway for the various values of the response factor to the relative velocity: (a) pattern for the coexisting phase; (b) pattern for freely moving trafc as t! 1. Fig.5. Headway prole of the kink density wave for various values of the response factor to the relative velocity and the kinkanti-kink density wave appears t = 10 4 (N = 100; x 0 = 5:0).

7 114 Xue Yu Vol. 11 Figure 4 shows the typical trafc patterns after a sufciently long time t = The spacetime evolution of the headway for various values of the response factor to the relative velocity has different properties. With increasing response factor to the relative velocity, any initial disturbances decay and any initial trafc flow with a non-uniform density prole evolves to a uniformly trafc flow. The pattern in Fig.4(a) exhibits the spacetime evolution of the headway for the coexisting phase after t = The symmetric kink anti-kink density wave is embodied as the trafc jam in the unstable state. The pattern in Fig.4(b) exhibits the spacetime evolution of the headway for the freely moving phase after t = 10 4, which satises the stability criterion V With increasing time, the uniform trafc flow with low density appears. Also, we can nd the headway prole of the kink density wave at t = Figure 5 is the headway prole of the kinkanti-kink density wave and the kink density wave appears. The amplitude of the density wave will decrease with increasing response factor. All the results show that the jamming transition occurs at a higher density and the effect of the relative velocity can stabilize the trafc flow and increase the delay time. This gives the vehicle enough time to adjust its velocity to the optimal state. [8] 6. Summary We extend the optimal velocity model of trafc to take into account the relative velocity. We analyse the stability and density waves of trafc flow with the perturbation method. The stability criterion is derived by the linear stability analysis. In the evolution of trafc flow from initial non-uniform to uniform distribution, the density wave is described through the Burgers equation or diffusion equation. With an increasing density of trafc flow, the density wave, respectively, is described by the KdV and mkdv equations in the regions of metastability and instability. At the critical point, the disturbance leads to a trafc jam. We carry out a simulation to compare the simulation result to the analytical result and we illustrate the evolution procedures of the trafc density wave. The amplitude of the density wave will decrease with increasing response factor to the relative velocity. All the results show that the jamming transition occurs at a higher density and the effect of the relative velocity can stabilize the trafc flow. eferences [1] Chowdhury D, Santen L and Schreckenberg A 000 Phys. ep [] Bando M, Hasebe K, Nakayama A, Shibata A and Sugiyama Y 1995 Phys. ev. E [] Kerner B S and Konhμauser P 199 Phys. ev. E 48 5 [4] Kurtz D A and Hong D C 199 Phys. ev. E 5 18 [5] Komatasu T and Sasa S 1995 Phys. ev. E [6] Muramatsu M and Nagatani T 1999 Phy. ev. E [7] Nagatani T 1999 Phys. ev. E [8] Treiber M et al 000 Phys. ev. E [9] Xue Y, Yun D L, Yuan Y W and Dai S Q 00 Commun. Theor. Phys. 8 0 [10] Xue Y, Yun DL,YuanYWand Dai S Q 00 Acta Phys. Sin (in Chinese) [11] Barlovic, Santen L, Schadschneider A and Schreckenberg M 1998 Eur. Phys. J. B 5 79