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

TL;DR

The paper develops a semi-implicit, asymptotic preserving finite-volume scheme on a MAC grid for the isentropic Euler equations with a congestion pressure p_ε(ρ) that enforces the constraint 0 ≤ ρ < 1. By introducing a velocity shift and leveraging a discrete energy framework, the scheme achieves positivity of ρ, energy stability, and a discrete density bound uniformly in ε. As ε → 0, the scheme accurately captures the free-congested limit, where congested regions (ρ = 1) interact with compressible regions (ρ < 1), thereby exhibiting the AP property. Numerical experiments in 1D and 2D validate AP behavior, preserve the density constraint, and demonstrate robustness across congested and low-density regimes with convergence toward the pressureless limit when appropriate. Overall, the method provides a reliable computational tool for simulating congested flow while remaining stable and accurate in the singular limit.

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.

Figures (12)

• Figure 1: The re-scaled pressure $p_\varepsilon(\varrho_\varepsilon)$ in \ref{['eqn:cong-pres']} for $\gamma = 2$ and different values of $\varepsilon$.
• Figure 2: Arrangement of the variables in one-dimensional grid
• Figure 3: Arrangement of the variables for a two-dimensional MAC discretization.
• Figure 4: Comparison of the numerical solutions versus the exact solutions with initial data \ref{['eqn:eg-1']} for different values of $\varepsilon$. Density (top) and momentum (bottom).
• Figure 5: Comparison of the numerical solutions versus the exact solutions with initial data \ref{['eqn:eg-2']} for $\varepsilon = 10^{-4}$. Density (left) and momentum (right).

Theorems & Definitions (23)

• Proposition 2.1: A priori energy estimates
• Definition 2.2: Entropy weak solutions
• Theorem 2.3: Existence of global weak solutions
• Remark 2.4
• Proposition 2.5: A priori estimates of the modified system
• Proposition 3.1: Discrete energy identities
• proof
• Theorem 3.2: Energy stability
• proof
• Theorem 3.3: Global energy estimate