# Unraveling the origin and identification of oscillatory modes in numerical low Mach number flow solutions

Jonathan Jung, Ibtissem Lannabi, Vincent Perrier

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Jonathan Jung, Ibtissem Lannabi, Vincent Perrier. Unraveling the origin and identification of oscillatory modes in numerical low Mach number flow solutions. ECCOMAS 2024 - The 9th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2024, Lisbon, Portugal. hal-04884331

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Submitted on 13 Jan 2025

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# Unraveling the origin and identification of oscillatory modes in numerical low Mach number flow solutions

Jonathan Jung<fnref n="1" />, Ibtissem Lannabi<fnref n="2" />,* and Vincent Perrier<fnref n="3" />

**Keywords:** Finite volume methods, low Mach number flows, compressible Euler equations, oscillatory modes, linear wave system.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Long time numerical solution to the linear wave system: Exact velocity field (left), numerical velocity field (middle), spurious mode isolated (right).</figcaption></figure>

Numerical methods for the compressible Euler equations may encounter difficulties with the zero Mach number limit. For instance, it is well-known that the Roe scheme lacks accuracy when dealing with low Mach number flows, especially on Cartesian grids [3]. In response to this issue, numerous modifications to the Roe scheme have been proposed. Many of these modifications fall into the category of Roe-pressure-centered schemes [4, 2]. Unfortunately, these schemes introduce other issues, such as the appearance of oscillatory modes in the numerical solution [1].

Leaning on an asymptotic analysis of the Euler equations as the Mach number goes to zero leads to a link with the long-time behavior of a linear wave system. Similarly, at the discrete level, Roe-pressure-centered schemes are known to be asymptotic consistent with a discretization of a linear wave system, in which a centered discretization of the pressure gradient is used. This discretization is referred to as the pressure-centered scheme.

<footnote n="1">¹ CNRS/Univ Pau and Pays Adour/E2S UPPA, Laboratoire de Mathématiques et de leurs Applications de Pau - Fédération IPRA, UMR5142 64000, Pau, France & Cagire team, Inria Bordeaux Sud-Ouest, France, jonathan.jung@univ-pau.fr</footnote>

<footnote n="2">² CNRS/Univ Pau and Pays Adour/E2S UPPA, Laboratoire de Mathématiques et de leurs Applications de Pau - Fédération IPRA, UMR5142 64000, Pau, France & Cagire team, Inria Bordeaux Sud-Ouest, France, ibtissem.lannabi@univ-pau.fr</footnote>

<footnote n="3">³ Cagire team, Inria Bordeaux Sud-Ouest, France & CNRS/Univ Pau and Pays Adour/E2S UPPA, Laboratoire de Mathématiques et de leurs Applications de Pau - Fédération IPRA, UMR5142 64000, Pau, France, vincent.perrier@inria.fr</footnote>
Due to the mentioned link, the issue of spurious modes arises also in the solution of the pressure-centered scheme, as illustrated in Figure 1. In this talk, we focus on the pressure-centered scheme, we identify the origin of the oscillatory modes problem and characterize these spurious modes. Furthermore, numerical simulations reveal that these modes significantly compromise the convergence of the numerical solution towards the expected one.

## REFERENCES

[1] S. Dellacherie. Checkerboard modes and wave equation. In *Proceedings of ALGO-RITMY*, volume 2009, pages 71–80, 2009.

[2] S. Dellacherie, J. Jung, P. Omnes, and P.-A. Raviart. Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system. *Mathematical Models and Methods in Applied Sciences*, Nov. 2016.

[3] H. Guillard and C. Viozat. On the behaviour of upwind schemes in the low Mach number limit. *Computers & fluids*, 28(1):63–86, 1999.

[4] F. Rieper. A low-Mach number fix for Roes approximate Riemann solver. *Journal of Computational Physics*, 230(13):5263–5287, 2011.