This work is devoted to the numerical simulation of low Mach number flows, modeled by the compressible Euler system. Commonly used solvers for discretizing this model are Godunov-type schemes. These schemes exhibit poor performance at low Mach number in terms of efficiency and accuracy.

Indeed, regarding the accuracy problem observed with quadrangular grids, it arises from the fact that the discrete solution fails to converge to the incompressible solution as the Mach number tends to zero. To overcome this accuracy problem, many fixes have been developed and consist in modifying the numerical diffusion of the original scheme. These corrections improve the accuracy of compressible schemes as the Mach number goes to zero. Unfortunately they introduce other problems, such as the appearance of numerical oscillations (checkerboard modes on a Cartesian grid) in the numerical solution, or the damping of acoustic waves as the Mach number goes to zero. Efficiency is also compromised as these schemes are stable under a more restrictive CFL condition compared to the original scheme.

In this talk, we propose to study the phenomenon of oscillations that plagues some of the fixes proposed in the literature. We focus on Roe-type fixes, in particular those that reduce the numerical diffusion on the jump of the normal velocity. The asymptotic analysis of these schemes leads to a discretization of a wave system in which the pressure gradient is centered. To better understand the phenomenon, we focus on the linear wave system. We then show that this fix is not TVD, unlike the Godunov scheme, which explains the appearance of numerical oscillations in the unsteady solution. Next, we study the long-time behavior of the numerical solution. It turns out that spurious stationary oscillations appear on the velocity field, preventing mesh convergence of the numerical solution. Moreover, the dimension of the space of spurious elements is huge, making it difficult to consider any form of filtering for these modes. We conclude this work by noting that the modification of the numerical diffusion makes it difficult to develop numerical methods that are both stable and accurate for low Mach number flows.