Calculus of Variations and Geometric Measure Theory

H. Leclerc - Q. Mérigot - F. Santambrogio - F. Stra

Lagrangian discretization of crowd motion and linear diffusion

created by santambro on 22 Jun 2019
modified by stra on 07 Jun 2021


Accepted Paper

Inserted: 22 jun 2019
Last Updated: 7 jun 2021

Journal: SIAM J. Num. An.
Year: 2020

ArXiv: 1905.08507 PDF
Links: .pdf file on hal


We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attractionrepulsion, 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.