Discrete and continuum links to a nonlinear coupled transport problem of interacting populations

TL;DR

This paper investigates microscopic models of interacting populations and their connection to nonlinear coupled transport equations, with applications to pedestrian dynamics and cross-diffusion phenomena, using entropy, Lyapunov functionals, and numerical methods.

Contribution

It introduces a framework linking particle systems to continuum coupled transport models with variational structures, including new analytical and numerical approaches.

Findings

Established a hydrodynamic limit for the particle systems.

Constructed explicit solutions to the nonlinear transport equations.

Developed numerical schemes for 2D continuum models.

Abstract

We are interested in exploring interacting particle systems that can be seen as microscopic models for a particular structure of coupled transport flux arising when different populations are jointly evolving. The scenarios we have in mind are inspired by the dynamics of pedestrian flows in open spaces and are intimately connected to cross-diffusion and thermo-diffusion problems holding a variational structure. The tools we use include a suitable structure of the relative entropy controlling TV-norms, the construction of Lyapunov functionals and particular closed-form solutions to nonlinear transport equations, a hydrodynamics limiting procedure due to Philipowski, as well as the construction of numerical approximates to both the continuum limit problem in 2D and to the original interacting particle systems.