Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Abstract

In this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales in both cases of a single road and road networks. We are especially interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. By means of both theoretical and numerical investigations, we show that, on a single road, the notion of Wasserstein distance fully catches the human perception of distance independently of the scale, while in the case of networks it partially loses its nice properties. KEYWORDS earth mover's distance, follow-the-leader model, LWR model, many-particle limit, multi-path model, networks, traffic flow, Wasserstein distance INTRODUCTIONIn this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales. More precisely, we are interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. We will consider both cases of a single road and road networks.Connections between microscopic (agent-based) and macroscopic (fluid dynamics) traffic flow models are already well established. Aw et al, 1 Greenberg, 2 and Di Francesco et al 3 investigated the many-particle limit in the framework of second-order traffic models, deriving the macroscopic Aw-Rascle-Zhang (ARZ) 4,5 model from a particular second-order microscopic Follow-the-Leader (FtL) 6,7 model. On the other hand, Colombo and Rossi, 8 Rossi, 9 Di Francesco and Rosini, 10 and Di Francesco et al 11 investigated the many-particle limit in the framework of first-order traffic models, deriving the Ligthill-Whitham-Richards (LWR) 12,13 model as the limit of a specific first-order FtL model. Let us also mention the papers by Forcadel et al 14 and Forcadel and Salazar,15 which investigate the many-particle limit exploiting the link between conservation laws and Hamilton-Jacobi equations.Moving to road networks, analogous connections are rarer. This is probably because of the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this context let us mention the paper by Cristiani and Sahu, 16 which investigates the many-particle limit of a first-order FtL model suitably extended to a road network. The corresponding macroscopic model appears to be the extension of the LWR model on networks introduced by Hilliges and Weidlich 17 and then extensively studied by Briani and Cristiani 18 and Bretti et al. 19 This paper focuses in particular on the comparison of solutions to the equations associated to first-order traffic models....

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

“…In the case D = 0, we can repeat the same steps as in Section 3.1 to find the local Maxwellian (16). Plugging it into (25) with ϕ(v) = 1, v, and recalling furthermore the interaction rule (7), we find the following second order macroscopic model:…”

“…The procedure followed in Section 4.2 to derive the Aw-Rascle model from the microscopic interactions (7) can be fruitfully exploited to obtain classes of second order macroscopic traffic models complying with the Aw-Rascle condition. Let us consider the interaction rules (1) with I given by (4) and…”

“…We consider now a space non-homogeneous scenario and we report results for the Boltzmann-type description of traffic flow coming from (10) under the binary interaction model (7) along with the quasi-invariant scaling (13). We start by giving the details of the discretisation technique.…”

“…let the selected pairs interact and, for each of them, set V n+1 k = V k , V n+1 h = V h, * according to the interaction rule (7) with the scaling (13) 6:…”

“…e.g. [7,22,36], and second order models, which include also an equation for the conservation or the balance of the mean speed, cf. e.g.…”