Mixed systems: ODEs – Balance laws

Borsche, Raul; Colombo, Rinaldo M.; Garavello, Mauro

doi:10.1016/j.jde.2011.08.051

Cited by 47 publications

(46 citation statements)

References 14 publications

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“…Our choice of Carathéodory solutions for (11) is justified by the particular bus velocity defined by (2). With this choice it is not possible for the bus to end up trapped in a queue unless its speed is equal to V b , in which case ω(ρ(t, y(t)+)) = ω(ρ(t, y(t)−)) = V b .…”

Section: The Cauchy Problem: Existence Of Solutionsmentioning

confidence: 99%

“…Other macroscopic models for moving bottlenecks in road traffic were recently proposed in [2,18]. Compared to those approaches, the model described here offers a more realistic definition of the slower vehicle speed, and a description of its impact on traffic conditions which is simpler to handle both analytically and numerically.…”

Section: Introductionmentioning

confidence: 99%

“…On one side, we deal with a strong coupling with the PDE and the ODE affecting each other, unlike [7,10], where the PDE solution does not depend on the ODE. On the other side, even if the ODE has discontinuous right-hand side, the particular definition of the model allows us to consider classical Carathéodory solutions, as in [3,2,4,7], instead of the weaker Filippov's generalized solutions needed in [10,18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

Monache¹,

Goatin²

2014

Journal of Differential Equations

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We consider a strongly coupled PDE-ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle is given by an ODE depending on the downstream traffic density. The moving constraint is expressed by an inequality on the flux, which models the bottleneck created in the road by the presence of the slower vehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.

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Section: The Cauchy Problem: Existence Of Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

Monache¹,

Goatin²

2014

Journal of Differential Equations

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“…The position of the particle is denoted by h(t) and its motion is governed by the Newton law where we have modeled the force of the fluid on the particle by λ (u(t, h(t)) − h ′ (t)), with λ > 0. For more details on this model, we refer to [13] and also to [2] (see also [4] for related models).…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model

Andreïanov¹,

Lagoutìère²,

Seguin³

et al. 2014

SIAM J. Math. Anal.

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The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl

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“…The traffic evolution is described with a non-linear scalar hyperbolic partial differential equation (PDE). In the last decade, several authors proposed different models to track the movement of a single car or a single vehicle among traffic, see [2,5,8,16] and references therein. In all these works the dynamics of the single vehicle is described by an ordinary differential equation (ODE).…”

Section: Introductionmentioning

confidence: 99%

Modeling of moving bottlenecks in traffic flow: a PDE-ODE model with moving density constraints

Monache

2014

ESAIM: ProcS

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Abstract. In this article we study a strongly coupled PDE-ODE problem from an analytical and numerical point of view. This model describes a moving bottleneck that can be produced by the presence of a bus on a road network. The PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, is intended in Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. We give an existence result and introduce a finite volume scheme. The algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are presented, in order to describe the behavior of the solution. Résumé. Dans cet article nousétudions un modèle couplé EDP-EDO d'un point de vue analytiqueet numérique et nous vérifions le caractère bien posé de ce problème. Le modèle aété introduit pour décrire un goulot d'étranglement mobile. En particulier, il peut décrire le mouvement d'un bus, qui rouleà une vitesse inférieureà celle des autres voitures, réduisant la capacité de la route et générant un rétrécissement mobile de la chausée. L'EDP est une loi de conservation scalaire avec une contrainte mobile sur la densité et l'EDO décrive la trajectoire du bus. Nous présentons un résultat d'existence des solutions du modèle, obtenu par la méthode d'approximation de suivi des fronts, et nous montrons des simulations numériques obtenues avec une méthode "front-capturing".

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Mixed systems: ODEs – Balance laws

Cited by 47 publications

References 14 publications

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model

Modeling of moving bottlenecks in traffic flow: a PDE-ODE model with moving density constraints

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