A conditionally linearly stable second-order traffic model derived from a Vlasov kinetic description

Romain Billot; Christophe Chalons; Florian De Vuyst; Nour-Eddin El Faouzi; Jacques Sau

doi:10.1016/j.crme.2010.07.018

A conditionally linearly stable second-order traffic model derived from a Vlasov kinetic description
[Un modèle de trafic du second ordre conditionnellement linéairement stable]

Présenté par : Évariste Sanchez-Palencia

Romain Billot ^1,² ; Christophe Chalons ² ; Florian De Vuyst ^2,^3,⁴ ; Nour-Eddin El Faouzi ¹ ; Jacques Sau ⁵

¹ Laboratoire d'ingénierie circulation transports (LICIT), INRETS-ENTPE, 25, avenue François-Mitterrand, case 24, 69675 Bron cedex, France
² École centrale Paris, laboratoire MAS, grande voie des vignes, 92295 Châtenay-Malabry cedex, France
³ Department of Mechanical Engineering, Energy Conversion Research Center, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, Kyoto 610-0321, Japan
⁴ École normale supérieure de Cachan, centre de mathématiques de leurs applications, 94235 Cachan cedex, France
⁵ LMFA UMR 5509, université Lyon 1, 69622 Villeurbanne cedex, France

Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 529-537.

Résumé
Abstract

Dans cette Note, un modèle de trafic du second ordre est construit à partir d'une description cinétique de type Vlasov. La partie homogène de ce système est hyperbolique. Néanmoins, en utilisant un terme source de relaxation de vitesse, le système complet non homogène s'avère être conditionnellement linéairement stable avec une région d'instabilité localisée dans le régime dense. La condition de stabilité linéaire dépend du choix du terme source et des paramètres ouverts du modèle. Les expériences numériques confirment l'analyse théorique. Pour une certaine classe de termes de source, le système est inconditionnellement linéairement stable ; les expérimentations numériques montrent l'apparition d'instabilités non linéaires qui évoluent en ondes « stop-and-go » dans la région de trafic dense.

A new second-order traffic model is derived from a nonlinear Vlasov type equation with a source term. The homogeneous part of the system is proven to be hyperbolic. Using a vehicle speed relaxation source term the full system appears to be conditionally linearly stable with instabilities in the dense traffic region. The stability condition depends on the choice of the source term and the model parameters. Numerical experiments confirm the analysis. For a class of source terms, the system is unconditionally linearly stable but numerical experiments show the appearance of nonlinear instabilities that evolve into stop-and-go waves in the dense region.

Reçu le : 2010-05-26
Accepté le : 2010-07-26
Publié le : 2010-08-23

DOI : 10.1016/j.crme.2010.07.018

Keywords: Dynamical systems, Traffic model, Vlasov equation, Hyperbolic equation, Linear stability, Stop-and-go wave, Numerical scheme
Mot clés : Systèmes dynamiques, Modèle de trafic, Équation cinétique, Équation hyperbolique, Stabilité linéaire, Stop-and-go, Schéma numérique

Affiliations des auteurs :

Romain Billot ^{1, 2} ; Christophe Chalons ² ; Florian De Vuyst ^{2, 3, 4} ; Nour-Eddin El Faouzi ¹ ; Jacques Sau ⁵

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     author = {Romain Billot and Christophe Chalons and Florian De Vuyst and Nour-Eddin El Faouzi and Jacques Sau},
     title = {A conditionally linearly stable second-order traffic model derived from a {Vlasov} kinetic description},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {529--537},
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     volume = {338},
     number = {9},
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     doi = {10.1016/j.crme.2010.07.018},
     language = {en},
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Romain Billot; Christophe Chalons; Florian De Vuyst; Nour-Eddin El Faouzi; Jacques Sau. A conditionally linearly stable second-order traffic model derived from a Vlasov kinetic description. Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 529-537. doi : 10.1016/j.crme.2010.07.018. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.018/

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