Stability Analysis of Optimal Velocity Model for Traffic and Granular Flow under Open Boundary Condition

Abstract

We analyzed the stability of the uniform flow solution in the optimal velocity model for traffic and granular flow under the open boundary condition. It was demonstrated that, even within the linearly unstable region, there is a parameter region where the uniform solution is stable against a localized perturbation. We also found an oscillatory solution in the linearly unstable region and its period is not commensurate with the periodicity of the car index space. The oscillatory solution has some features in common with the synchronized flow observed in real traffic.KEYWORDS: optimal velocity model, traffic flow, granular pipe flow, open boundary condition, stability of uniform solutionThe traffic flow on a highway 1, 3, 2, 4, 5) and the granular pipe flow 6,7,8,9,10) are similar in a number of characteristics. Both consist of discrete elements following dissipative dynamics, both are quasi one-dimensional systems, and spontaneous density waves with the power law occur in both systems. To understand the common structure of such phenomena, both discrete and continuous models have been suggested and have succeeded in reproducing some aspects of the spatiotemporal patterns. 8,11,12,13,14,15,16,17) In recent years, a car-following model, termed the "optimal velocity model" (OV model), was proposed. 11, 12)The model is realistic enough to be able to reproduce a spontaneous traffic jam as in real traffic, but simple enough to be regarded as a model of granular pipe flow without much modification. In fact, a model similar to the OV model has been proposed as a granular flow model and it has been demonstrated to exhibit power law behavior in the power spectra of the density wave in certain situations. 8)Even though a number of analyses of the OV model have been performed, most of them were conducted under the periodic boundary condition (PBC).11, 12, 13) The PBC corresponds to the situation that cars run in a circuit, but it is not suitable for most situations in real traffic observations. When we consider the situation of granular pipe flow, the PBC is quite unrealistic. Furthermore, from the experimental results of the granular pipe flow, it has been suggested that the condition of the downstream boundary is important. 9)In the present work, we investigate the OV model under the open boundary condition (OBC), and examine the effects of the boundary condition on the stability of the system.The OV model is a dynamical model for a onedimensional system which consists of discrete elements; the elements can be cars or grain particles, but they tend * E-mail: namiko@stat.phys.kyushu-u.ac.jp * * E-mail: naka4scp@mbox.nc.kyushu-u.ac.jp to run at the speed determined by a local configuration. When the (n + 1)th car precedes the nth car (Fig. 1) and the position of the nth car at the time t is denoted by x n (t), then the dynamics is governed by the equation of motionẍwhere b n (t) represents the headway of the nth car defined byand the dot denotes the time derivative. Here, a is a sensitivity constant and U (b) is the optimal v...

Statistical physics of vehicular traffic and some related systems

“…An alternative possibility has been explored in recent works based on the carfollowing approach [89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. This formulation is based on the assumption that V desired n depends on the distance-headway of the n-th vehicle, i.e., V desired n (t) = V opt (∆x n (t)) so thatẍ…”

Macroscopic traffic models from microscopic car-following models

“…Motivated by this observation, we transform the leading order term of Σ in Eq. (13) in such a way that it preserves the same long wavelength behavior but suppresses fluctuations in short wavelength components with k ≫ ρ h . To implement this idea, one first notes that Eq.…”

“…There are empirical indications of multiple dynamic phases in the traffic flow and dynamic phase transitions [5][6][7][8][9]. Several theoretical explanations [10][11][12][13][14][15] for the empirical results were suggested. Also physical phenomena such as self-organized criticality and hysteresis [16] were revealed.…”

Intermittent unstable structures induced by incessant constant disturbances in the full velocity difference car-following model

“…To learn the behavior of traffic flow, various traffic flow models have been proposed and studied: carfollowing models [2][3][4][5][6][7][8][9][10][11][12][13][14], cellular automaton models [15,16], gas-kinetic models [17][18][19][20][21] and fluid-dynamic models [22][23][24][25][26][27]. These models are successfully applied to the computer simulation of traffic, and the physical phenomena such as the nonlinear waves and nonequilibrium phase transitions are revealed, which are consistent with the experimental data [28].…”

“…Recently, Mitarai and Nakanishi have analyzed the stability of traffic flow under open boundary condition [13,14]. They pointed out that for the linearly unstable traffic, two different situations were distinguished, i.e., convective instability and absolute instability.…”