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A convergent staggered scheme for the variable density incompressible Navier-Stokes equations

Authors: J. C. Latché and K. Saleh
Journal: Math. Comp. 87 (2018), 581-632
MSC (2010): Primary 35Q30, 35Q55, 65N12, 76M10, 76M12
Published electronically: August 7, 2017
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Abstract: In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem ( $ \mathrm {L}^\infty $-estimate for the density, $ \mathrm {L}^\infty (\mathrm {L}^2)$- and $ \mathrm {L}^2(\mathrm {H}^1)$-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.

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Additional Information

J. C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France

K. Saleh
Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 bd 11 novembre 1918, F-69622 Villeurbanne cedex, France

Keywords: Variable density, Navier-Stokes equations, staggered schemes, analysis, convergence
Received by editor(s): February 24, 2014
Received by editor(s) in revised form: February 21, 2015, February 29, 2016, and October 28, 2016
Published electronically: August 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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