An approximation scheme for a Hamilton–Jacobi equation defined on a network

Camilli, Fabio; Festa, Adriano; Schieborn, Dirk

doi:10.1016/j.apnum.2013.05.003

Cited by 19 publications

(22 citation statements)

References 10 publications

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“…Viscosity solutions for HJ equations on networks have been introduced in recent years (see e.g. [12], [24], [3]), the case of the directed graph -where the arcs can be traveled only in one direction -can be considered a sub-case of the latter.…”

Section: Strategy Modeling On the Networkmentioning

confidence: 99%

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

Festa

Goatin

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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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.

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Section: Strategy Modeling On the Networkmentioning

confidence: 99%

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

Festa

Goatin

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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“…Apart from [14] mentioned above, we are only aware of two other works. A convergent semi-Lagrangian scheme is introduced in [8] for equations of eikonal type. In [21], an adapted Lax-Friedrichs scheme is used to solve a traffic model; it is worth mentioning that this discretization implies to pass from the scalar conservation law to the associated Hamilton-Jacobi equation at each time step.…”

Section: Related Resultsmentioning

confidence: 99%

Error estimates for a finite difference scheme associated with Hamilton–Jacobi equations on a junction

Guerand

Koumaiha

2019

Numer. Math.

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This paper is concerned with monotone (time-explicit) finite difference schemes associated with first order Hamilton-Jacobi equations posed on a junction. They extend the schemes recently introduced by Costeseque, Lebacque and Monneau (2013) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive optimal error estimates of order (∆x) 1 2 in L ∞ loc for junction conditions of optimal-control type at least if the flux is "strictly limited".

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“…We also refer to [7], [15] for different numerical discretizations of conservation law on networks in the framework of vehicular traffic motion. Concerning the Eikonal equation (2), we recall that approximations of Hamilton-Jacobi equations on networks are discussed in [11] for finite differences and in [10] for semi-Lagrangian schemes. The system (9) has been introduced as a discretization of the continuous problem (1)-(2), but nevertheless it inherits some dynamical properties of the original model and it can be interpreted as a discrete-time finite state model for the flow of pedestrians on a graph in the following way.…”

Section: The Discrete Hughes' Modelmentioning

confidence: 99%

A discrete Hughes model for pedestrian flow on graphs

Camilli¹,

Festa²,

Tozza³

2017

Networks &Amp; Heterogeneous Media

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In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.

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An approximation scheme for a Hamilton–Jacobi equation defined on a network

Cited by 19 publications

References 10 publications

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

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

Error estimates for a finite difference scheme associated with Hamilton–Jacobi equations on a junction

A discrete Hughes model for pedestrian flow on graphs

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