Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams
Florent Berthelin Thierry Goudon Bastien Polizzi Magali Ribot
Networks & Heterogeneous Media 2017, 12(4): 591-617 doi: 10.3934/nhm.2017024

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

keywords: Traffic flows unilateral constraints finite volume schemes asymptotic preserving schemes
A model for the evolution of traffic jams in multi-lane
Florent Berthelin Damien Broizat
Kinetic & Related Models 2012, 5(4): 697-728 doi: 10.3934/krm.2012.5.697
In [8], Berthelin, Degond, Delitala and Rascle introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model consists of a Pressureless Gas Dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. In the present paper we propose an improvement of this model by allowing the road to be multi-lane piecewise. The idea is to use the maximal constraint to model the number of lanes. We also add in the model a parameter $\alpha$ which model the various speed limitations according to the number of lanes. We present the dynamical behaviour of clusters (traffic jams) and by approximation with such solutions, we obtain an existence result of weak solutions for any initial data.
keywords: Traffic flow models weak solutions constrained pressureless gas dynamics traffic jams. multi-lane
The multi-water-bag equations for collisionless kinetic modeling
Nicolas Besse Florent Berthelin Yann Brenier Pierre Bertrand
Kinetic & Related Models 2009, 2(1): 39-80 doi: 10.3934/krm.2009.2.39
In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. After recalling the link of the multi-water-bag model with kinetic formulation of conservation laws, we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. These models are very promising because they reveal to be very useful for the theory and numerical simulations of laser-plasma and gyrokinetic physics. In this paper we prove some existence and uniqueness results for classical solutions of these different models. We next propose numerical schemes based on Discontinuous Garlerkin methods to solve these equations. We then present some numerical simulations of non linear problems arising in plasma physics for which we know some analytical results.
keywords: hyperbolic systems of conservation laws plasma physics. water bag model collisionless kinetic equations discontinuous Galerkin methods Cauchy problem

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