Numerical simulations of the Euler system with congestion constraint

TL;DR

This paper develops and tests asymptotic preserving numerical schemes for simulating the Euler system with a congestion constraint, effectively capturing complex transitions in density dynamics.

Contribution

It adapts two AP schemes originally for low Mach number limits to handle the congestion constraint in Euler systems, enabling accurate simulations of dynamic transitions.

Findings

Schemes successfully simulate density transitions in 1D and 2D cases.

Numerical tests demonstrate the schemes' robustness and accuracy.

Effective handling of asymptotic dynamics transitions in congested flows.

Abstract

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes.