From the modelling of driver's behavior to hydrodynamic models and problems of traffic flow

Bellomo, Nicola; Marasco, Addolorata; Romano, António

doi:10.1016/s1468-1218(01)00032-3

Cited by 42 publications

(15 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Models based on the kinetic theory of gases have also been developed in the literature, see [40,51,50,4]. However, such an approach is also, in our opinion, as inappropriate as the continuum approach for the reasons discussed later.…”

Section: Models Based On the Kinetic Theory Of Gasesmentioning

confidence: 99%

“…This must be done steadily and the concentration, as defined by them, will never be dilute 4 , especially if the speeds of the cars are bounded (and this is particularly so nowadays with strict speed limits inside the city limits as well as on the highways). The only way to achieve the criterion of "no interaction between cars" is to move the car out of the section of interest and this can be achieved only if the car can travel at infinite or at the very least unacceptably high speeds or 4 We also do not believe that the notion of concentration as defined in such an approach is a meaningful concept. In fact, we believe that it is not meaningful to develop a homogenized field theory (a system of partial differential equations in the "flow domain," the section of a highway).…”

Section: Models Based On the Kinetic Theory Of Gasesmentioning

confidence: 99%

See 1 more Smart Citation

A review of the mathematical models for traffic flow

Tyagi

Darbha

Rajagopal

2009

Int J Adv Eng Sci Appl Math

View full text Add to dashboard Cite

show abstract

Section: Models Based On the Kinetic Theory Of Gasesmentioning

confidence: 99%

Section: Models Based On the Kinetic Theory Of Gasesmentioning

confidence: 99%

A review of the mathematical models for traffic flow

Tyagi

Darbha

Rajagopal

2009

Int J Adv Eng Sci Appl Math

View full text Add to dashboard Cite

show abstract

“…In traffic modelling, this leads to so-called second-order traffic models, which include elements such as anticipation length and reaction time. For the more involved physics of non-kinematic multiphase flow models, we refer again to [39][40][41][42][43]; for traffic models, see [44][45][46][47].…”

Section: Limitations Of Kinematic Modelsmentioning

confidence: 99%

“…To obtain the last inequality, we have used the second CFL condition appearing in (46). Thus, (48) is valid, and the proof is complete for each interior point.…”

Section: Theorem 32 Consider Scheme 4 Defined Bymentioning

confidence: 99%

See 1 more Smart Citation

A family of numerical schemes for kinematic flows with discontinuous flux

et al. 2007

View full text Add to dashboard Cite

Multiphase flows of suspensions and emulsions are frequently approximated by spatially one-dimensional kinematic models, in which the velocity of each species of the disperse phase is an explicitly given function of the vector of concentrations of all species. The continuity equations for all species then form a system of conservation laws which describes spatial segregation and the creation of areas of different composition. This class of models also includes multi-class traffic flow, where vehicles belong to different classes according to their preferential velocities. Recently, these models were extended to fluxes that depend discontinuously on the spatial coordinate, which appear in clarifier-thickener models, in duct flows with abruptly varying cross-sectional area, and in traffic flow with variable road surface conditions. This paper presents a new family of numerical schemes for such kinematic flows with a discontinuous flux. It is shown how a very simple scheme for the scalar case, which is adapted to the "concentration times velocity" structure of the flux, can be extended to kinematic models with phase velocities that change sign, flows with two or more species (the system case), and discontinuous fluxes. In addition, a MUSCL-type upgrade in combination with a Runge-Kutta-type time discretization can be devised to attain second-order accuracy. It is proved that two particular schemes within the family, which apply to systems of conservation laws, preserve an invariant region of admissible concentration vectors, provided that all velocities have the same sign. Moreover, for the relevant case of a multiplicative flux discontinuity and a constant maximum density, it is proved that one scalar version converges to a BV t entropy solution of the model. In the latter case, the compactness proof involves -R. Bürger et al. a novel uniform but local estimate of the spatial total variation of the approximate solutions. Numerical examples illustrate the performance of all variants within the new family of schemes, including applications to problems of sedimentation, traffic flow, and the settling of oil-in-water emulsions.

show abstract

On the modelling of vehicular traffic and crowds by kinetic theory of active particles

Bellomo

Bellouquid²

2010

Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences

View full text Add to dashboard Cite

From the modelling of driver's behavior to hydrodynamic models and problems of traffic flow

Cited by 42 publications

References 32 publications

A review of the mathematical models for traffic flow

A review of the mathematical models for traffic flow

A family of numerical schemes for kinematic flows with discontinuous flux

On the modelling of vehicular traffic and crowds by kinetic theory of active particles

Contact Info

Product

Resources

About