NHM
A continuum-discrete model for supply chains dynamics
Gabriella Bretti Ciro D’Apice Rosanna Manzo Benedetto Piccoli
This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [10], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [15].
keywords: networks conservation laws fluid-dynamic models finite difference schemes Supply chains
DCDS-S
An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments
Gabriella Bretti Maya Briani Emiliano Cristiani
In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence ad hoc external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.
keywords: LWR model traffic flow multipopulation model Godunov scheme multipath model networks.
NHM
Numerical approximations of a traffic flow model on networks
Gabriella Bretti Roberto Natalini Benedetto Piccoli
We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occur and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.
keywords: finite difference schemes boundary conditions. traffic flow fluid-dynamic models Scalar conservation laws
DCDS-B
Fast algorithms for the approximation of a traffic flow model on networks
Gabriella Bretti Roberto Natalini Benedetto Piccoli
New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced. Numerical results and efficiency tests are presented in order to show the behaviour of FSF in comparison with G, FG and a conservative scheme of second order.
keywords: fluid-dynamic models Scalar conservation laws traffic flow finite difference schemes.

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