Homogenization of a microscopic pedestrians model on a convergent junction

¹ Lebanese American University, Department of computer science and mathematics, Byblos campus, P.O. Box 36, Byblos, Lebanon
² Normandie Univ, INSA de Rouen, LMI (EA 3226 – FR CNRS 3335), 76000 Rouen, France
³ 685 Avenue de 1’Université, 76801 St Etienne du Rouvray cedex, France

^* Corresponding author: nader.elkhatib@lau.edu.lb

Received: 24 January 2022
Accepted: 3 June 2022

Abstract

In this paper, we establish a rigorous connection between a microscopic and a macroscopic pedestrians model on a convergent junction. At the microscopic level, we consider a “follow the leader” model far from the junction point and we assume that a rule to enter the junction point is imposed. At the macroscopic level, we obtain the Hamilton-Jacobi equation with a flux limiter condition at x = 0 introduced in Imbert and Monneau [Ann. Sci. l’École Normale Supér. 50 (2017) 357-414], To obtain our result, we inject using the “cumulative distribution functions” the microscopic model into a non-local PDE. Then, we show that the viscosity solution of the non-local PDE converges locally uniformly towards the solution of the macroscopic one.