Self-sustained nonlinear waves in traffic flow

In analogy to gas-dynamical detonation waves, which consist of a shock with an attached exothermic reaction zone, we consider herein nonlinear traveling wave solutions to the hyperbolic (“inviscid”) continuum traffic equations. Generic existence criteria are examined in the context of the Lax entropy conditions. Our analysis naturally precludes traveling wave solutions for which the shocks travel downstream more rapidly than individual vehicles. Consistent with recent experimental observations from a periodic roadway [Y. Sugiyama et al., N. J. Phys. 10, 033001 (2008)], our numerical calculations show that nonlinear traveling waves are attracting solutions, with the time evolution of the system converging toward a wave-dominated configuration. Theoretical principles are elucidated by considering examples of traffic flow on open and closed roadways.

Figure 1

Characteristics on both sides of a left (a) and right (b) shock in the frame of the shock. The flow direction is from left to right in (a) and right to left in (b). The equations of the characteristics are $C_{\pm }:d\eta /dt=\left(u-s\pm c\right)/\tau$, where $\eta$ is defined by Eq. (2.1), and $s$ is the shock speed.

Figure 2

(a) Typical profiles for $\tilde{u}/{\tilde{u}}_0$ (solid curve) and $u/{\tilde{u}}_0$ (dashed curve) as functions of $\rho /{\rho }_M$, where $u=s+m/\rho$. Equivalently, these profiles may be plotted against $u/{\tilde{u}}_0$ by employing transformation (2.2), as shown in (b)—the dashed curve is then just a line with unit slope. The two curves will either not intersect at all, be tangent at a single point, or have two transversal intersections. The case of interest to us is the one with two transversal intersections, as depicted here: $u_2$ is the intersection with a smaller velocity and a larger density ${\rho }_2=m/\left(u_2-s\right)$—solid circles, while the other intersection (open circles) defines $u_1$ and ${\rho }_1$. In order to satisfy the conditions in Eq. (2.11), the sonic point must coincide with $\left(u_2,{\rho }_2\right)$. Finally, a physically meaningful solution requires $u>0$ everywhere. Hence, $u_2>0$ is needed. As shown by panel (b), this condition is equivalent to the statement: when $\tilde{u}=0$, that is $\rho ={\rho }_M$, the corresponding $u=u_M=s+m/{\rho }_M$ is positive.

Figure 3

[(a) and (b)] Comparison of theoretical (thick curves) and numerical (thin curves) solutions. The numerical solutions are the final asymptotic state of an evolution started with a small perturbation of a uniform unstable base state. Numerical results are either nominally inviscid or are obtained selecting ${\Gamma }_3=1.5,2.5,5.0,10,20$, where ${\Gamma }_3\equiv \tau {\rho }_M\mu$. Arrows show the direction of increasing traffic viscosity $\mu$ [see Eq. (5.15)]. The panels show profiles of $\rho /{\rho }_M$ and $u/{\tilde{u}}_0$ versus $\eta \tau {\rho }_M=\left(x-st\right){\rho }_M$. In the latter case, numerical results are omitted for clarity. Equations and parameters are as in Eqs. (5.19, 5.20, 5.21), with $\lambda =46\ell =230\text{m}$. In (a), $\mathcal{N}=22$ mimicking the experiment of Sugiyama et al. [14]. In (b), $\mathcal{N}=16$. Stars indicate the sonic point. Corresponding representative vehicle trajectories, reconstructed from the theoretical solution, are also shown in (c) and (d) where, in both cases, vehicles move from left to right.

Figure 4

Parameters characterizing the exact solutions in Sec. 5A, for a fixed road length $\lambda =46\ell =230\text{m}$, as functions of the number of vehicles, $\mathcal{N}$, or, equivalently, the average nondimensional traffic density ${\rho }_{avg}/{\rho }_M=\mathcal{N}/\left(\lambda {\rho }_M\right)$. Specifically, curves of ${\rho }^{-}/{\rho }_M$, ${\rho }^{+}/{\rho }_M$, $u^{-}/{\tilde{u}}_0$, and $u^{+}/{\tilde{u}}_0$ are shown, in addition to $r_2/{\tilde{u}}_0$ — as given in Eq. (4.7). The vertical dashed lines specify the linear stability boundaries ${\rho }_{avg}/{\rho }_M=0.016$ and ${\rho }_{avg}/{\rho }_M=0.984$ indicated by Eq. (5.12). The equations and parameters are as in Eqs. (5.19, 5.20, 5.21). Note that $r_2$ goes through $0$ precisely at the stability boundaries whereupon $\left[\rho \right]=0$, $\left[u\right]=0$.

Figure 5

(a) Theoretical profiles of the traffic density through a jamiton on an infinite road with various ${\rho }_{\pm \infty }$. As with Fig. 3, stars denote the sonic point. (b) Representative vehicle trajectories for ${\rho }_{\pm \infty }/{\rho }_M=0.35$.

Figure 6

(Color online) Measured vehicle trajectories due to Treiterer. The straight line shows the experimental measurement of Sugiyama et al. [14] (figure reproduced from Sugiyama et al. [14] with permission).

Figure 7

(Color online) Multiple jamitons on a long (nonperiodic) road. Light and dark curves show, respectively, the theoretical and nominally inviscid numerical solutions. The difference is visible only in the expanded inset, which also presents a representative distribution of individual vehicles along the bottom. Here, for technical reasons, we select ${\Gamma }_1=20$ and ${\Gamma }_2=0.5$. Qualitatively identical results are obtained with ${\Gamma }_1\simeq 8.0$ and ${\Gamma }_2\simeq 0.13$.