An Asymptotic Preserving and Energy Stable Scheme for the Euler System with Congestion Constraint

TL;DR

This paper introduces an asymptotic preserving and energy stable finite volume scheme for the Euler system with congestion constraints, effectively capturing the transition from compressible to free-congested flow as the small parameter tends to zero.

Contribution

The authors develop a novel semi-implicit scheme that is positivity preserving, energy stable, and accurately captures the limiting behavior of the Euler system with congestion pressure.

Findings

The scheme is positivity preserving and energy stable.

Numerical results confirm the scheme's ability to capture both compressible and congested dynamics.

The scheme satisfies a discrete density constraint.

Abstract

In this work, we design and analyze an asymptotic preserving (AP), semi-implicit finite volume scheme for the scaled compressible isentropic Euler system with a singular pressure law known as the congestion pressure law. The congestion pressure law imposes a maximal density constraint of the form $0\leq \varrho <1$, and the scaling introduces a small parameter $\varepsilon$ in order to control the stiffness of the density constraint. As $\varepsilon\to 0$, the solutions of the compressible system converge to solutions of the so-called free-congested Euler equations that couples compressible and incompressible dynamics. We show that the proposed scheme is positivity preserving and energy stable. In addition, we also show that the numerical densities satisfy a discrete variant of the constraint. By means of extensive numerical case studies, we verify the efficacy of the scheme and show that the scheme is able to capture the two dynamics in the limiting regime, thereby proving the AP property.