Carlotta Donadello

Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(⋅), we are able to model the capacity drop phenomenon at the exit. Existence and stability results for the Cauchy problem with Lipschitz constraint function p(⋅) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(⋅). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretizati...

A second-order model for vehicular traffics with local point constraints on the flow

In this paper we present a second-order model based on the Aw, Rascle, Zhang model (ARZ) for vehicular traffics subject to point constraints on the flow, its motivation being, for instance, the modeling of traffic lights along a road. We first introduce a definition of entropy solution by choosing a family of entropy pairs analogous to the Kruzhkov entropy pairs for scalar conservation laws; then we apply the wave-front tracking method to prove existence and a priori bounds for the entropy solutions of constrained Cauchy problem for ARZ with initial data of bounded variation and piecewise constant constraints. The case of solutions attaining values at the vacuum is considered. We construct an explicit example to describe some qualitative features of the solutions.

Riemann problems with non--local point constraints and capacity drop.

In the present note we discuss in details the Riemann problem for a one-dimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the well--posedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non--local operator depending on the solution itself. We provide several explicit examples. We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.

On the Attainable Set for a Scalar Nonconvex Conservation Law

We deal with a Cauchy problem for a scalar conservation law in one space dimension. The flux function is assumed to be nonconvex, in particular, to have a single inflection point. We consider a compactly supported initial datum and regard it as a control. The main result of the paper states sufficient conditions for a function

On the formation of scalar viscous shocks problem

v

An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

to be attained at a fixed time

Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws

T

Solutions of the Aw-Rascle-Zhang system with point constraints

by a trajectory of the conservation law.

Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks

Consider a viscous scalar conservation law with smooth, possibly non-convex flux. Assume that the (arbitrarily large) initial data remains in a small neighbourhood of given states u-, u+ as x → ∞, with u-, u+ connected by a stable shock profile. We then show that the solution eventually forms a viscous shock. The time needed for the shock to appear is the main focus of the present analysis.

Initial–boundary value problems for continuity equations with BV coefficients

In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called metastable phase. The model is given by the Lighthill–Whitham–Richards model in the free-flow phase and by the Aw–Rascle–Zhang model in the congested phase. In particular, we study the existence of solutions to Cauchy problems satisfying a local point constraint on the density flux. We prove that if the constraint F is higher than the minimal flux