# Lagrangian discretization of crowd motion and linear diffusion

## TL;DR

This paper introduces a Lagrangian discretization method for modeling crowd motion and linear diffusion, using optimal transport to enforce density constraints and demonstrating convergence and numerical implementation in 2D.

## Contribution

It presents a novel Lagrangian discretization scheme for crowd motion and diffusion equations, with convergence proofs and practical 2D numerical examples.

## Key findings

- Convergence of discrete measures to continuous PDE solutions in 1D.
- Feasibility of 2D numerical implementation.
- Effective enforcement of density constraints via optimal transport.

## Abstract

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.