Lagrangian discretization of crowd motion and linear diffusion

Hugo Leclerc (1) , Quentin Merigot (1) , Filippo Santambrogio (2) , Federico Stra (1)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 MMCS - Modélisation mathématique, calcul scientifique

We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal transport problem at each time step. We prove the convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one. In a second part, we show how a similar approach can be used to construct a Lagrangian discretization of a linear advection-diffusion equation, interpreted as a gradient flow in Wasserstein space. We provide also a numerical implementation in 2D to demonstrate the feasibility of the computations.

Domaines: Analyse numérique [math.NA]

Hugo Leclerc, Quentin Merigot, Filippo Santambrogio, Federico Stra. Lagrangian discretization of crowd motion and linear diffusion. 2019. ⟨hal-02124815v1⟩