General phase transition models for vehicular traffic with point constraints on the flow

Abstract

In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one-lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free-flow phase and by a system of 2 conservation laws in the congested phase. The free-flow phase is described by a one-dimensional fundamental diagram corresponding to a Newell-Daganzo type flux. The congestion phase is described by a two-dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced. KEYWORDS conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-Zhang model, vehicular traffic, unilateral point constraint, Riemann problem INTRODUCTIONThis paper deals with phase transition models (PT models for short) of hyperbolic conservation laws for vehicular traffic. More precisely, we focus on macroscopic models that describe vehicular traffic along a unidirectional one-lane road, which has neither entrances nor exits and where overtaking is not allowed.In the specialized literature, vehicular traffic is shown to behave differently depending on whether it is free or congested. This leads to consider 2 different regimes corresponding to a free-flow phase (denoted by Ω f ) and a congested phase (denoted by Ω c ). The PT models analysed here are given by a scalar conservation law in the free-flow phase, coupled with a 2 × 2 system of conservation laws in the congested phase. The coupling is achieved via phase transitions, namely, discontinuities that separate 2 states belonging to different phases and that satisfy the Rankine-Hugoniot condition.This two-phase approach was introduced in Colombo 1 and is motivated by experimental observations; according to which for low densities, the flow of vehicles is free and approximable by a one-dimensional flux function, while at high densities, the flow is congested and covers a two-dimensional domain in the fundamental diagram, see Colombo 1, Figure 1.1 or Blandin et al. 2, Figure 3.1 Hence, it is reasonable to describe the dynamics in the free regime by a first-order model (a scalar PDE) and those in the congested regime by a second-order model (a 2 × 2 system of PDEs).In literature, 1 Colombo proposed to let the dynamics in free-flow phase Ω f be governed by the classical LWR model by Lighthill, Whitham, and Richards 3,4 ; on the other hand, the congested phase Ω c includes one more equation for the conservation of a linearized momentum. We recall that the LWR model expresses the conservation of the number of vehicles and assumes that the velocity is a function of the density alone. Furthermore, Colombo's model uses a Greenshields type (strictly parabolic) flux function in the free-flow regime and one consequence is that Ω f cannot intersect Ω c , see Colombo. 1, Remark 2Math Meth Appl Sci. 2017;40:6623-6641.wile...

An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

“…For this reason in (7) (and then also in (8)) we consider test functions φ such that φ(·, 0) ≡ 0. This is in the same spirit of the solutions considered in [6,[12][13][14][15] for traffic through locations with reduced capacity. However, with this choice for the test functions in (7) and (8) we loose the possibility to better characterize the (density) flux at x = 0 associated to non-classical shocks.…”

“…Then also T n is left continuous in time. By the monotonicity of w →v(w), w →ŵ(w), v →v(v), v →w(v), see Remark 2.3, and the definitions ofΥ andΥ given in (12), we have that Υ n (t) = TV + v w n (t, ·) ; (−∞, 0) + TV − ŵ w n (t, ·) ; (−∞, 0)…”

“…These two-phase models have been recently generalized in [6,12] by considering Riemann problems, namely Cauchy problems for piecewise constant initial data with a single jump, coupled with a constraint on the density flux, so that at the interface x = 0 the density flux of the solution must be lower than a given constant quantity F . This condition is referred to as unilateral point constraint and can be thought of as a pointwise bottleneck at x = 0 that hinders the density flow, see [20] and the references therein.…”

“…In the present paper, we focus on the constrained version proposed in [12] for the model introduced in [5] and prove an existence result for constrained Cauchy problems. More precisely, we use the Riemann solvers established in [5] and [12] in a wave-front tracking scheme and prove that the obtained approximate solution u n converges (up to a subsequence) to a globally defined solution of the constrained Cauchy problem with general BV-initial data, at least in the case F ≥ f − c , where the threshold value f − c is the minimal density flux of the metastable phase, see Figure 1. At last, in the case F < f − c we give sufficient conditions on the initial data that ensure the convergence of u n to a globally defined solution of the constrained Cauchy problem.…”

Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic

“…In the present paper, we focus on the constrained Cauchy problem without a metastable phase. More precisely, as in [7] we use the Riemann solvers established in [8] and [20] in a wave-front tracking scheme to construct a sequence of approximate solutions {u n } n , which converges at the limit n → ∞ to a solution u. We show that the total variation (in space) of u n becomes arbitrarily large as n goes to infinity and that the total variation of solution u blows up in finite time, even if the initial datum u o has bounded total variation.…”