# Low Mach number limit of some staggered schemes for compressible barotropic flows

## TL;DR

This paper analyzes the low Mach number limit of staggered discretization schemes for barotropic Navier-Stokes equations, proving their asymptotic preserving property and convergence to incompressible flow solutions.

## Contribution

It provides a rigorous proof that three different staggered schemes are asymptotic preserving as Mach number approaches zero, ensuring accurate incompressible flow simulation.

## Key findings

- All three schemes are asymptotic preserving.
- Convergence to incompressible flow solutions is rigorously established.
- Stability conditions depend on the discretization of the convection term.

## Abstract

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.

---
Source: https://tomesphere.com/paper/1803.09568