This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier-Stokes equations.

翻译：本文研究了两种针对可压缩斯托克斯方程速度-密度公式的混合间断伽辽金（HDG）离散格式，重点关注若干期望的结构性质：可证明的收敛性、密度非负性与质量约束的保持，以及梯度鲁棒性。后者在静力平衡（即压力梯度平衡重力力的情形）等均衡态场景中可显著提升精度。其中一种格式采用H(div)相容的速度试探函数空间，确保满足所有上述性质；而全间断方法虽能满足除梯度鲁棒性外的所有条件。本文还提出了两种格式的高阶版本，并通过三个数值基准问题进行了对比分析。最后一个算例表明，该梯度鲁棒性对可压缩纳维-斯托克斯方程中非静力平衡态的准确模拟同样具有重要意义。