Well‐posedness of IBVP for 1D scalar non‐local conservation laws

We consider the initial boundary value problem (IBVP) for a non‐local scalar conservation law in one space dimension. The non‐local operator in the flux function is not a mere convolution product, but it is assumed to be aware of boundaries. Introducing an adapted Lax‐Friedrichs algorithm, we provide various estimates on the approximate solutions that allow to prove the existence of solutions to the original IBVP. The uniqueness follows from the Lipschitz continuous dependence on initial and boundary data, which is proved exploiting results available for the local IBV

Modeling the impact of on-line navigation devices in traffic flows

We consider a macroscopic multi-population traffic flow model on networks accounting for the presence of drivers (or autonomous vehicles) using navigation devices to minimize their instantaneous travel cost to destination. The strategic choices of each population differ in the degree of information about the system: while part of the agents knows only the structure of the network and minimizes the traveled distance, others are informed of the current traffic distribution, and can minimize their travel time avoiding the most congested areas. In particular, the different route choices are computed solving eikonal equations on the road network and they are implemented at road junctions. The impact on traffic flow efficiency is illustrated by numerical experiments. We show that, even if the use of routing devices contributes to alleviate congestion on the whole network, it also results in increased traffic on secondary roads. Moreover, the generalized use of real-time information can even deteriorate the efficiency of the network

Uniqueness and Stability of LinftyL^infty Solutions for Temple Class Systems with Boundary and Properties of the Attainable Sets

Summary: The authors consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws on the quarter t-x plane where $t,x geq 0$. For a class of initial and boundary data in $L^infty$ with possibly unbounded variation, they construct a flow of solutions that depend continuously, in the $L^1$ distance, both on the initial data and on the boundary data. Moreover, we show that each trajectory of such flow provides the unique weak solution of the corresponding initial-boundary value problem that satisfies an entropy condition of Oleinik type. Next, they study the initial-boundary value problem from the point of view of control theory, taking the initial data fixed and considering, in connection with a prescribed set of boundary data regarded as admissible controls, the set of attainable profiles at a fixed time $T>0$ and at a fixed point $x>0$, establishing closure and compactness of such sets in the $L^1$ topolog..

A multilane macroscopic traffic flow model for simple networks

We prove the well-posedness of a system of balance laws inspired by [H. Holden and N. H. Risebro, SIAM J. Math. Anal., 51 (2019), pp. 3694-3713], describing macroscopically the traffic flow on a multilane road network. Motivated by real applications, we allow us for the the presence of space discontinuities both in the speed law and in the number of lanes. This allows us to describe a number of realistic situations. Existence of solutions follows from compactness results on a sequence of Godunov's approximations, while L\bfone -stability is obtained by the doubling of variables technique. Some numerical simulations illustrate the behavior of solutions in sample cases

Comparative study of macroscopic traffic flow models at road junctions

We qualitatively compare the solutions of a multilane model with those produced by the classical Lighthill-Whitham-Richards equation with suitable coupling conditions at simple road junctions. The numerical simulations are based on the Godunov and upwind schemes. Several tests illustrate the models' behaviour in different realistic situations

The initial–boundary value problem for general non-local scalar conservation laws in one space dimension

We prove a global well-posedness result for a class of weak entropy solutions of bounded variation (BV) of scalar conservation laws with non-local flux on bounded domains, under suitable regularity assumptions on the flux function. In particular, existence is obtained by proving the convergence of an adapted Lax–Friedrichs algorithm. Lipschitz continuous dependence from initial and boundary data is derived applying Kružhkov’s doubling of variable technique

Non-local multi-class traffic flow models

We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform L1 and BV estimates for the sequence of approximate solutions, locally in time. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition

Numerical study of macroscopic pedestrian flow models

We analyze numerically macroscopic models of crowd dynamics: classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost function standing for minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-element method to solve the problems and present error analysis in case of the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit. In particular, we analyze the effect on the total evacuation time of the level of compression, the desired speed and the presence of obstacles.On analyse plusieurs modèles numériques pour la dynamique des foules : le modèle classique de Hughes et le modèle du second ordre, qui est une extension au mouvement de piétons du modèle de Payne-Whitham pour le trafic routier. La direction du mouvement est obtenue par résolution d'une équation Eikonale avec une fonction de coût dépendant de la densité, de manière à minimiser le temps de déplacement et éviter les endroits congestionnés. On utilise une méthode mixte éléments / volumes finis pour résoudre le problème et on présente une analyse d'erreur pour la résolution de l'équation Eikonale, le calcul de gradient et le modèle du second ordre, aboutissant à une précision du premier ordre. On montre que le modèle de Hughes est incapable de reproduire la dynamique des foules complexes, comme les ondes de type "stop-and-go" et le phénomène de colmatage apparaissant aux goulots d'étranglement. Enfin, avec le modèle du second ordre, on étudie numériquement l'évacuation de piétons d'une salle avec une sortie étroite. An particulier, on analyse l'effet du niveau de compression, de la vitesse de déplacement et la présence d'obstacle sur le temps total d'évacuation

The initial–boundary value problem for general non-local scalar conservation laws in one space dimension

We prove a global well-posedness result for a class of weak entropy solutions of bounded variation (BV) of scalar conservation laws with non-local flux on bounded domains, under suitable regularity assumptions on the flux function. In particular, existence is obtained by proving the convergence of an adapted Lax–Friedrichs algorithm. Lipschitz continuous dependence from initial and boundary data is derived applying Kružhkov’s doubling of variable technique