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Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations


Authors: T. Gallouët, R. Herbin, J.-C. Latché and D. Maltese
Journal: Math. Comp. 87 (2018), 1127-1163
MSC (2010): Primary 35Q30, 65N12, 76N10, 76N15, 65M12
DOI: https://doi.org/10.1090/mcom/3260
Published electronically: September 19, 2017
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Abstract: We prove in this paper the convergence of the marker and cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two- or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.


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Additional Information

T. Gallouët
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
Email: thierry.gallouet@univ-amu.fr

R. Herbin
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
Email: raphaele.herbin@univ-amu.fr

J.-C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France
Email: jean-claude.latche@irsn.fr

D. Maltese
Affiliation: IMATH, Université du Sud Toulon-Var, BP 20132 - 83957 La Garde Cedex, France
Email: david.maltese@univ-amu.fr

DOI: https://doi.org/10.1090/mcom/3260
Keywords: Compressible fluids, Navier-Stokes equations, Cartesian grids, marker and cell scheme, convergence
Received by editor(s): July 6, 2016
Received by editor(s) in revised form: December 6, 2016
Published electronically: September 19, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society