0 Ju n 19 98 Structure and Instability of High-Density Equations for Traffic Flow Dirk Helbing

Similar to the treatment of dense gases, fluid-dynamic equations for the dynamics of congested vehicular traffic are derived from Enskog-like kinetic equations. These contain additional terms due to the anisotropic vehicle interactions. The calculations are carried out up to Navier-Stokes order. A linear instability analysis indicates an additional kind of instability compared to previous macroscopic traffic models. The relevance for describing granular flows is outlined. 51.10. +y,47.50. +d,47.55. -t,89.40. +k Typeset using REVTEX 1 An efficient infrastructure is an essential precondition for every industrialized country. Therefore, the considerable deterioration of the traffic situation on ‘freeways’ during the last decade is a serious problem. Not only does impeded traffic cause economic losses of many billions of dollars each year, it also produces serious ecological damages. Thus, great efforts have been made to develop methods for traffic optimization and forecasts, for which reliable traffic simulations are necessary. Consequently, research on traffic dynamics has recently become a very important topic. During the last years, numerous results have been published on microscopic models [1–4], including cellular automata models [2] and molecular-dynamics-like models [3,4], kinetic models [4–10], and macroscopic (fluid-dynamiclike) models [11,4–10], aiming at an understanding of stop and go traffic. The topic is related to the fields of non-linear dynamics [11], phase transitions [1–4,11], and stochastic processes [1,2]. Macroscopic traffic models are not only suitable for on-line simulations of traffic networks, but also for analytical investigations. Some very recent publications proposed to derive realistic traffic equations from the ‘microscopic’ dynamics of driver-vehicle units via a kinetic approach [4–10]. However, for the following reasons, these attempts have been only partly successful. Either the models treat the vehicles like point-like objects [5–7]. Then, the resulting macroscopic models are only valid for free traffic flow. Or the models take into account the finite space requirements of vehicles [4,8–10]. However, the calculations were carried out up to Euler order only, based on the (zeroth order) approximation of local equilibrium. This assumes that the form of the equilibrium velocity distribution remains unchanged in dynamic situations, but it is given by the local values of the density ρ(r, t), average velocity V (r, t), etc. It will be demonstrated that corresponding traffic models are not even valid in linear approximation, since their linear stability analysis gives totally misleading results. To solve this problem, we must calculate the (first order) Navier-Stokes corrections of the macroscopic traffic equations, which take into account the structural change of the velocity distribution in inhomogeneous traffic situations. For ordinary gases, the Navier-Stokes terms (transport terms) are calculated from the 2 kinetic equation by means of the Chapman-Enskog method [12]. An approximate, but more intuitive method bases on the relaxation time approximation [13,14] (see below). For kinetic traffic models the situation is more involved, since the interaction term does not vanish in equilibrium situations. Now, this problem has been solved. In the following, we will show how to derive realistic traffic equations which include corrections due to vehicular space requirements and Navier-Stokes terms. Since these non-trivial corrections change the structure of the equations, they cause an additional instability. For the same reason, the low-density regime does not allow an extrapolation to situations at high densities. The presented method is also relevant for understanding instabilities in granular flows, since granular collisions are also not energy conserving. Recently, a lot of publications tried to tackle these interesting problems with fluid-dynamic equations derived from the associated kinetic equation [15–17]. However, most of them are restricted to Euler order or the low-density regime [16,17], thereby neglecting relevant sources of stability and instability. Probably for this reason, these approaches have not been fully successful in describing the formation of density waves in sand which is falling through a vertical pipe [17,18]. The model: Since the number of vehicles on a (for simplicity: circular) freeway is conserved, the kinetic traffic equation for the phase space density ρ̃(r, v, t) of vehicles with velocity v = dtr at place r and time t has the form of a continuity equation with a sink/source term: ∂tρ̃ + ∂r(ρ̃v) + ∂v(ρ̃ dtv) = (∂tρ̃)ss . (1) As usual, we will assume the acceleration law dtv = (V0 − v)/τ , where τ denotes a densitydependent acceleration time and V0 the desired velocity, which is assumed to be the same for all vehicles, here (case of a speed limit). The sink/source term (∂tρ̃)ss originates from sudden (non-differentiable) velocity changes. It splits up into a velocity-diffusion term due to fluctuations of the acceleration behavior (‘imperfect driving’) and an interaction term: (∂tρ̃)ss = ∂ 2 v(ρ̃D) + (∂tρ̃)int . (2)