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## Fundamental diagrams for kinetic equations of traffic flow

Venue: | Discrete Contin. Dyn. Syst. Ser. S |

Citations: | 3 - 2 self |

### Citations

176 |
A simplified theory of kinematic waves in highway traffic, II: Queueing at freeway bottlenecks.
- NEWELL
- 1993
(Show Context)
Citation Context ...nd possibly nonlinearly, to zero. In particular, with two speed classes only, namely v1 = 0 and v2 = Vmax, the resulting fundamental diagram is the triangular one introduced by Daganzo [7] and Newell =-=[15]-=- as a simplification of more elaborated diagrams for both theoretical and numerical purposes, see also [2]. Moreover, it is interesting to observe that the speed diagram is in per8 fect agreement with... |

84 |
The Physics of Traffic
- Kerner, L
- 2004
(Show Context)
Citation Context ... comprised between 0 and ρmax. The flow regime for 0 ≤ ρ ≤ σ is called free, whereas that for σ < ρ ≤ ρmax is called congested. Fundamental relationships are either measured experimentally, cf. e.g., =-=[4, 13]-=-, or expressed analytically, cf. e.g., [11, 16]. See also [14], where a systematic study is developed aiming at an analytical characterization of fundamental diagrams out of experimental data. Analyti... |

37 |
Hyperbolic phase transitions in traffic flow.
- Colombo
- 2002
(Show Context)
Citation Context ...tion is given as an external input to the model itself. To date, this seems to be impossible in a purely macroscopic approach, where the phase transition has to be postulated heuristically, cf. e.g., =-=[3, 5]-=-. 3 Asymptotic analysis and equilibria In this section we study the qualitative properties of system (10), particularly its asymptotic trends and equilibria which, as stated in Section 2, are at the b... |

30 | On the mathematical theory of vehicular traffic flow I. Fluid dynamic and kinetic
- Bellomo, Delitala, et al.
- 2002
(Show Context)
Citation Context ...sional gas-kinetic theory, the interaction rate ηhk may depend on the speeds of the interacting particles (we recall that e.g., in the discrete Boltzmann equation one has ηhk ∝ |vk − vh| but see also =-=[6]-=-). However, considering that cars do not interact through collisions and that more or less frequent interactions are essentially due to the level of traffic congestion on the road, we assume that the ... |

19 | Bayen A.: ‘A general phase transition model for vehicular traffic
- Blandin, Work, et al.
- 2011
(Show Context)
Citation Context ...tion is given as an external input to the model itself. To date, this seems to be impossible in a purely macroscopic approach, where the phase transition has to be postulated heuristically, cf. e.g., =-=[3, 5]-=-. 3 Asymptotic analysis and equilibria In this section we study the qualitative properties of system (10), particularly its asymptotic trends and equilibria which, as stated in Section 2, are at the b... |

11 |
2005b. A variational formulation of kinematic waves: Solution methods
- Daganzo
(Show Context)
Citation Context ...onotonically, and possibly nonlinearly, to zero. In particular, with two speed classes only, namely v1 = 0 and v2 = Vmax, the resulting fundamental diagram is the triangular one introduced by Daganzo =-=[7]-=- and Newell [15] as a simplification of more elaborated diagrams for both theoretical and numerical purposes, see also [2]. Moreover, it is interesting to observe that the speed diagram is in per8 fec... |

10 |
M.: Modelling and simulation of vehicular traffic jam formation
- Degond, Delitala
- 2008
(Show Context)
Citation Context ...laborated diagrams for both theoretical and numerical purposes, see also [2]. Moreover, it is interesting to observe that the speed diagram is in per8 fect agreement with that obtained numerically in =-=[8]-=- by simulating a microscopic “follow-the-leader” model. This confirms practically that the kinetic approach successfully retains the microscopic character of car-to-car interactions, although the repr... |

10 |
Mathematical modeling of vehicular traffic: a discrete kinetic theory approach
- Delitala, Tosin
(Show Context)
Citation Context ...behaviors of cars, we refrain from arriving at the detail of the microscopic scale and settle instead at the mesoscopic (or kinetic) level. Particularly, we refer to some recently introduced 2 models =-=[1, 9, 10]-=- in which the space of microstates of cars, namely the spatial position x and the speed v, is partly or fully discrete in order to incorporate in the kinetic representation the intrinsic microscopic g... |

8 | Numerical simulations of traffic data via fluid dynamic approach,” - Blandin, Bretti, et al. - 2009 |

7 |
Towards the modeling of vehicular traffic as a complex system: a kinetic theory approach
- Bellouquid, Angelis, et al.
(Show Context)
Citation Context ...behaviors of cars, we refrain from arriving at the detail of the microscopic scale and settle instead at the mesoscopic (or kinetic) level. Particularly, we refer to some recently introduced 2 models =-=[1, 9, 10]-=- in which the space of microstates of cars, namely the spatial position x and the speed v, is partly or fully discrete in order to incorporate in the kinetic representation the intrinsic microscopic g... |

7 | A fully-discrete-state kinetic theory approach to traffic flow on road networks - Fermo, Tosin - 2015 |

7 |
Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations
- Günther, Klar, et al.
(Show Context)
Citation Context ...0, ρmax] system (6) admits a unique stable equilibrium then these mappings are actual functions; otherwise, they define multivalued diagrams, which have also been studied in the literature, cf. e.g., =-=[12]-=-. 2.1 A prototypical case study In order to substantiate more the issues set forth above, we now specialize Eq. (10) by exemplifying a structure of the interaction rate and of the table of games. We p... |

7 |
Vehicular traffic: a review of continuum mathematical models
- Piccoli, Tosin
- 2009
(Show Context)
Citation Context ... for 0 ≤ ρ ≤ σ is called free, whereas that for σ < ρ ≤ ρmax is called congested. Fundamental relationships are either measured experimentally, cf. e.g., [4, 13], or expressed analytically, cf. e.g., =-=[11, 16]-=-. See also [14], where a systematic study is developed aiming at an analytical characterization of fundamental diagrams out of experimental data. Analytical relationships are mostly used in the mathem... |

6 | Traffic Flow on Networks – Conservation Laws Models
- Garavello, Piccoli
- 2006
(Show Context)
Citation Context ... for 0 ≤ ρ ≤ σ is called free, whereas that for σ < ρ ≤ ρmax is called congested. Fundamental relationships are either measured experimentally, cf. e.g., [4, 13], or expressed analytically, cf. e.g., =-=[11, 16]-=-. See also [14], where a systematic study is developed aiming at an analytical characterization of fundamental diagrams out of experimental data. Analytical relationships are mostly used in the mathem... |

4 |
From experiments to hydrodynamics traffic flow models
- Bonzani, Mussone
- 2003
(Show Context)
Citation Context ... comprised between 0 and ρmax. The flow regime for 0 ≤ ρ ≤ σ is called free, whereas that for σ < ρ ≤ ρmax is called congested. Fundamental relationships are either measured experimentally, cf. e.g., =-=[4, 13]-=-, or expressed analytically, cf. e.g., [11, 16]. See also [14], where a systematic study is developed aiming at an analytical characterization of fundamental diagrams out of experimental data. Analyti... |

1 |
Fundamental diagram of traffic flow
- Li, Zhang
(Show Context)
Citation Context ...alled free, whereas that for σ < ρ ≤ ρmax is called congested. Fundamental relationships are either measured experimentally, cf. e.g., [4, 13], or expressed analytically, cf. e.g., [11, 16]. See also =-=[14]-=-, where a systematic study is developed aiming at an analytical characterization of fundamental diagrams out of experimental data. Analytical relationships are mostly used in the mathematical theory o... |