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.

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.