Self-Organized Hydrodynamics with congestion and path formation in crowds

Pierre Degond (1) , Jiale Hua (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 EDP - Equations aux Dérivées Partielles

A continuum model for self-organized dynamics is numerically investigated. The model describes systems of particles subject to alignment interaction and short-range repulsion. It consists of a non-conservative hyperbolic system for the density and velocity orientation. Short-range repulsion is included through a singular pressure which becomes infinite at the jamming density. The singular limit of infinite pressure stiffness leads to phase transitions from compressible to incompressible dynamics. The paper proposes an Asymptotic-Preserving scheme which takes care of the singular pressure while preventing the breakdown of the Courant-Friedrichs-Lewy (CFL) stability condition near congestion. It relies on a relaxation approximation of the system and an elliptic formulation of the pressure equation. Numerical simulations of impinging clusters show the efficiency of the scheme to treat congestions. A two-fluid variant of the model provides a model of path formation in crowds.

Mots clés: Self-propelled particles; orientation dynamics; self-organization; hydrodynamic limit; volume exclusion; congestion; jamming; finite volumes; Asymptotic-Preserving scheme; herds; crowds; path formation

Domaines: Physique mathématique [math-ph]

Pierre Degond, Jiale Hua. Self-Organized Hydrodynamics with congestion and path formation in crowds. Journal of Computational Physics, 2013, 237, pp.299-319. ⟨10.1016/j.jcp.2012.11.033⟩. ⟨hal-00717995⟩