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.

翻译：本文针对具有奇异压力定律（称为拥塞压力定律）的标度化可压缩等熵欧拉系统，设计并分析了一种渐近保持（AP）、半隐式的有限体积格式。拥塞压力定律施加了形式为$0\leq \varrho <1$的最大密度约束，而标度化引入了一个小参数$\varepsilon$以控制密度约束的刚性。当$\varepsilon\to 0$时，可压缩系统的解收敛于所谓的自由-拥塞欧拉方程的解，该方程耦合了可压缩与不可压缩动力学。我们证明了所提格式具有保正性与能量稳定性。此外，我们还证明了数值密度满足该约束的一个离散变体。通过大量的数值案例研究，我们验证了该格式的有效性，并表明该格式能够捕捉极限状态下的两种动力学行为，从而证明了其渐近保持性质。