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


Authors: R. Herbin, J.-C. Latché and K. Saleh
Journal: Math. Comp. 90 (2021), 1039-1087
MSC (2020): Primary 35Q31, 65N12, 76M10, 76M12
DOI: https://doi.org/10.1090/mcom/3604
Published electronically: February 4, 2021
MathSciNet review: 4232217
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Abstract: In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one, so that the resulting scheme is unconditionally stable, and explicit for the second one, so that the scheme is stable under a CFL condition involving the material velocity only. We prove rigorously 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 tends to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof of convergence of the solutions of the (continuous) barotropic Navier-Stokes equations to that of the incompressible Navier-Stokes equation as the Mach number vanishes. Numerical results performed with a hand-built analytical solution show the behaviour that is expected from the analysis. Additional numerical results are obtained for the shock solutions of problems which are not in the scope of the present non dimensionalization but are nevertheless interesting to understand the behaviour of the scheme.


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Additional Information

R. Herbin
Affiliation: Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
MR Author ID: 244425
ORCID: 0000-0003-0937-1900
Email: raphaele.herbin@univ-amu.fr

J.-C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SA21, Cadarche, St. Paul-lez Durance, 13115 France
MR Author ID: 715367
Email: jean-claude.latche@irsn.fr

K. Saleh
Affiliation: Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
MR Author ID: 960776
Email: saleh@math.univ-lyon1.fr

Keywords: Compressible Navier-Stokes equations, low Mach number flows, finite volumes, Crouzeix-Raviart scheme, Rannacher-Turek scheme, finite elements, staggered discretizations
Received by editor(s): January 31, 2019
Received by editor(s) in revised form: December 5, 2019, and July 22, 2020
Published electronically: February 4, 2021
Article copyright: © Copyright 2021 American Mathematical Society