# Finite Volume approximations of the Euler system with variable congestion

## TL;DR

This paper develops an asymptotic preserving finite volume scheme for simulating the Euler system with variable congestion, effectively modeling crowd dynamics with singular pressure effects in multiple dimensions.

## Contribution

It introduces a novel AP scheme for the Euler system with congestion, including a second order accurate version and validation in 1D and 2D scenarios.

## Key findings

- The scheme accurately captures crowd behavior in simulations.
- The second order scheme improves precision over first order.
- Numerical results demonstrate realistic crowd dynamics modeling.

## Abstract

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure. This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensional test-cases and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.