In this paper, we study the behaviour at low Mach number of numerical schemes based on staggered discretizations for the barotropic Navier-Stokes equations. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The last two schemes differ by the discretization of the convection term: linearly implicit for the first one, so the resulting scheme is unconditionnally stable, and explicit for the second one, so the scheme is stable under a CFL condition involving the material velocity only. We rigorously prove that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tend to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof already known in the continuous case.

中文翻译：

---

可压缩正压流动的一些交错方案的低马赫数限制

在本文中，我们研究了基于正压 Navier-Stokes 方程的交错离散化的低马赫数数值方案的行为。考虑了三种时间离散化：隐式时间方案和两种非迭代压力校正方案。后两个方案的区别在于对流项的离散化：第一个方案是线性隐式的，因此结果方案是无条件稳定的，而第二个方案是显式的，因此该方案在仅涉及材料速度的 CFL 条件下是稳定的。我们严格证明这三种变体在以下意义上是渐近保持的：对于给定的网格和给定的时间步长，用马赫数消失序列获得的一系列解趋于不可压缩流的标准方案的解。