arXiv subjects

Laurent Navoret

Publications and source records attributed to Laurent Navoret.

22 records · Page 2Linked to original sources

arXiv2013

Macroscopic models of collective motion and self-organization

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view of its possible extensions to other kinds of collective motion.

math-ph↗

arXiv2010

Numerical simulations of the Euler system with congestion constraint

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes.

math.NA↗

arXiv2009

Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior

The Vicsek model is a very popular individual based model which describes collective behavior among animal societies. A macroscopic version of the Vicsek model has been derived from a large scale limit of this individual based model \cite{degond2007ml}. In this work, we want to numerically validate this Macroscopic Vicsek model (MV). To this aim, we compare the simulations of the macroscopic and microscopic models one with each other. The MV model is a non-conservative hyperbolic equation with a geometric constraint. Due to the lack of theory for this kind of equations, we derive several equivalents for this system leading to specific numerical schemes. The numerical simulations reveal that the microscopic and macroscopic models are in good agreement provided that we choose one of the proposed formulations based on a relaxation of the geometric constraint. This confirms the relevance of the macroscopic equation but it also calls for a better theoretical understanding of this type of equations.

math.AP↗

arXiv2009

Congestion in a macroscopic model of self-driven particles modeling gregariousness

We analyze a macroscopic model with a maximal density constraint which describes short range repulsion in biological systems. This system aims at modeling finite-size particles which cannot overlap and repel each other when they are too close. The parts of the fluid where the maximal density is reached behave like incompressible fluids while lower density regions are compressible. This paper investigates the transition between the compressible and incompressible regions. To capture this transition, we study a one-dimensional Riemann problem and introduce a perturbation problem which regularizes the compressible-incompressible transition. Specific difficulties related to the non-conservativity of the problem are discussed.

math-ph↗