Non classical solution of a conservation law arising in vehicular traffic modelling

ESAIM: PROCEEDINGS AND SURVEYS, December 2016, Vol. 55, p. 131-147 Emmanuel FRÉNOD, Emmanuel MAITRE, Antoine ROUSSEAU, Stéphanie SALMON and Marcela SZOPOS Editors NON CLASSICAL SOLUTION OF A CONSERVATION LAW ARISING IN VEHICULAR TRAFFIC MODELLING ∗ Mathieu Fabre 1 , Sylvain Faure 2 , Mathieu Laurière 3 , Bertrand Maury 4 and Charlotte Perrin 5 Abstract. We are interested in this paper in the modelling and numerical simulation of some phenomena that are observed in the context of car dynamics, in particular the appearance of persistent jams upstream critical points with no real cause of flux limitation. We shall consider the case of a stable jam on a freeway upstream an accident that took place on the opposite lane. This situation is not properly handled by most models, either micro-or macroscopic ones, since it corresponds to a phenomenon that does not have a counterpart in gas dynamics, for which only entropy solution are usually considered as physically feasible. The approach we propose consists in accounting for the very behaviour of agents in the neighbourhood of the discontinuity, and makes it possible to numerically recover in a robust way steady traffic jams. Résumé. Nous nous intéressons dans cet article à la modélisation et à la simulation numérique de phénomènes particuliers observés dans le contexte du trafic routier, plus particulièrement le phénomène d’apparition et de persistance de bouchons en amont de points critiques sans cause objective de ralentissement. Nous considérerons notamment le cas d’un bouchon persistant sur une autoroute en amont d’un accident qui s’est produit sur la voie d’en face. Cette situation n’est pas reproduite par la plupart des modèles, qu’ils soient microscopiques ou macroscopiques, car elle correspond à un phénomène qui n’a pas d’équivalent en dynamique des gaz, pour lesquels seules les solutions dites entropiques sont en général considérées comme correspondant à un comportement observable dans la réalité. L’approche que nous proposons, aux niveaux microscopique et macroscopique, consiste à prendre en compte le comportement particulier des agents humains au voisinage de la discontinuité, et permet de retrouver numériquement de façon robuste des bouchons stables. Introduction Modelling traffic has become in the recent years a major challenge for the applied mathematics community. Among all the issues raised by vehicular traffic behaviours we are particularly interested in the modelling of congestion phenomenon, how traffic jams appear, persist and disappear. There are mainly three approaches to model traffic motion, each corresponding to a specific scale: microscopic scale (each entity is individually followed), macroscopic scale (the collection of entities ∗ Acknowledgements : The authors would like to thank F. Lagoutière for his fruitful suggestions. This project is supported by ANR Project Isotace (ANR-12-MONU-0013). 1 EPFL SB MATHICSE MNS (Bât. MA), station 8, CH 1015 Lausanne (Switzerland), 2 Laboratoire de Mathématiques d’Orsay, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France. 3 NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai (China), 4 LMO, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France & DMA, École Normale Supérieure, Paris 5 Institut für Mathematik, RWTH Aachen University, c EDP Sciences, SMAI 2017 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201655131 132 ESAIM: PROCEEDINGS AND SURVEYS is described by a local density), and mesoscopic scale (kinetic description). In this paper we will focus on the first two descriptions, the interested reader is referred to [4] for a review of the kinetic models. The most direct way to describe the motion of a collection of vehicles is to represent each of them individually and to follow the trajectory of each agent, taking into account all the interactions with the surrounding agents. In one-dimensional situations (a one-lane road or a corridor), a simple description of the dynamics is provided by Follow-The-Leader models used for instance in [5]– [12]. In this context congestion is handled by introducing short-range correction terms which mimic some repulsion when an agent gets closer to the agent next to him. This is the approach adopted in Follow-The-Leader models, where the velocity is supposed to decrease with the inter-agent distance, and is taken to be zero at a minimal distance corresponding to the size of the agents, which is not negligible in the discrete description. Note that such models are not based on classical mechanics principles: interaction terms are not symmetric, since each agent is influenced by the one that lies in front of him, with no reciprocity. Another mathematical description widely used consists in considering the assembly of vehicles as a continuous medium, and working with averaged variables: the average velocity of agents in an elementary volume (or length in the one-dimensional setting), and the density, which corresponds to the number of agents per unit volume. We then make an analogy with fluids by writing a balance equation expressing at the macroscopic level the conservation of the number of agents. If one wants to simulate the behaviour of a large number of agents, the macroscopic description may be attractive compared to the microscopic description for the computational cost and for the relative ease with which one can calibrate the model due to the low number of parameters. As we shall see, although such models are based on principles that are similar to the Follow-TheLeader approach, the “directionality of influence” that we previously mentioned will disappear in the expression of interaction terms. This very distinction between the two approaches will play a crucial role in the present paper. Intuitively, discrete models like the Follow-The-Leader models should converge towards continuous conservation laws. This issue has been the object of few and relatively recent mathematical works. In [3], the authors prove a mathematical link between the discrete solutions of a second order (in time) Follow-The-Leader model for car traffic, and the semi-discretization in time of a macroscopic solution to a system of conservation laws called Aw-Rascle model. The formal idea consists in letting the number of vehicles go to infinity while imposing a particular scaling in time and space which ensures that the density and the velocity remain fixed. Concerning first order Follow-The-Leader models, which neglect the effects of inertia in the motion, this question has been treated by Colombo and Rossi in [8] and by Di Francesco and Rosini in [11]. In the latter, the authors prove that the discrete solutions satisfy a discrete inequality of Oleinik’s type which yields strong controls of the semi-discrete density, and the strong convergence towards the unique entropy solution of the limit conservation law. This limit passage between (...truncated)