Free-congested and micro-macro descriptions of traffic flow
Francesca Marcellini
We present two frameworks for the description of traffic, both consisting in the coupling of systems of different types. First, we consider the Free--Congested model [7,11], where a scalar conservation law is coupled with a $2\times2$ system. Then, we present the coupling of a micro- and a macroscopic models, the former consisting in a system of ordinary differential equations and the latter in the usual LWR conservation law, see [10]. A comparison between the two different frameworks is also provided.
keywords: macroscopic traffic models microscopic traffic models. Traffic models hyperbolic systems of conservation laws ordinary differential equations
Coupling conditions for the $3\times 3$ Euler system
Rinaldo M. Colombo Francesca Marcellini
This paper is devoted to the extension to the full $3\times3$ Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.
keywords: Conservation Laws at Junctions Coupling Conditions at Junctions.
The Riemann Problem at a Junction for a Phase Transition Traffic Model
Mauro Garavello Francesca Marcellini

We extend the Phase Transition model for traffic proposed in [8], by Colombo, Marcellini, and Rascle to the network case. More precisely, we consider the Riemann problem for such a system at a general junction with $n$ incoming and $m$ outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model. For special junctions, we prove that the Riemann solver is well defined.

keywords: Phase transition model hyperbolic systems of conservation laws continuum traffic models Riemann problem Riemann solver
Existence of solutions to a boundary value problem for a phase transition traffic model
Francesca Marcellini

We consider the initial boundary value problem for the phase transition traffic model introduced in [9], which is a macroscopic model based on a 2×2 system of conservation laws. We prove existence of solutions by means of the wave-front tracking technique, provided the initial data and the boundary conditions have finite total variation.

keywords: Hyperbolic systems of conservation laws continuum traffic models wave-front tracking boundary value problem
Modeling and analysis of pooled stepped chutes
Graziano Guerra Michael Herty Francesca Marcellini
We consider a mathematical model describing pooled stepped chutes where the transport in each pooled step is described by the shallow-water equations. Such systems can be found for example at large dams in order to release overflowing water. We analyze the mathematical conditions coupling the flows between different chutes taken from the engineering literature. For the case of two canals divided by a weir, we present the solution to the Riemann problem for any initial data in the subcritical region, moreover we give a well-posedness result. We finally report on some numerical experiments.
keywords: Hyperbolic Conservation Laws on Networks Management of Water.
Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results
Rinaldo M. Colombo Francesca Marcellini Elena Rossi
This paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator -- prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation.
keywords: predatory -- prey systems. Nonlocal conservation laws parabolic-hyperbolic problems laser cutting modeling

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