Traffic Flow on a Road Network

Abstract

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem.Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

A Fluid Dynamic Model for Telecommunication Networks with Sources and Destinations

“…Moreover, again by subsequences, ρ ν converges weakly in L 1 loc . We then can complete the proof as in [6].…”

“…To pass to the limit one has to bound the number of waves and the BV norm of approximate solutions, see [5,6]. In the next section we prove a BV bound on the density for the case of junctions with two incoming and two outgoing transmission lines, for both the routing algorithms.…”

“…In this section we derive estimates on the total variation of the densities along a wave-front tracking approximate solution (constructed as in [6]) for the algorithm (RA2) with the traffic distribution function of type 2b). This allows to construct the solutions to the Cauchy problem in standard way, see [5].…”

“…A first model for telecommunication networks, similar to that introduced recently for car traffic, has been proposed in [9] where two algorithms for dynamics at nodes were considered and existence of solution to Cauchy Problems was proved. The idea is to follow the approach used in [11] for road networks (see also [6], [8], [10], [13], [15], [16], [17]), introducing sources and destinations in the telecommunication model described in [9] and thus taking care of the paths of the packets inside the network.…”

A domain decomposition method for conservation laws with discontinuous flux function

“…We present an alternative view on those problems by introducing a domain decomposition procedure [19,15,8]. Our approach is inspired from recent work on partial differential equation on network geometries, see for example [9,4,1], and therefore coupling conditions for the subdomains are similar to the coupling conditions at the vertices of a network model.…”