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, discretization of p(·) may induce non-uniqueness and instability of solutions.

Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

ROSINI, Massimiliano Daniele
2014

Abstract

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, discretization of p(·) may induce non-uniqueness and instability of solutions.
Andreianov, Boris; Donadello, Carlotta; Rosini, Massimiliano Daniele
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11392/2356654
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 24
  • ???jsp.display-item.citation.isi??? 20
social impact