An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations
The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 51-97.

In this paper we present a pressure correction scheme for the compressible Navier-Stokes equations. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured grids, or a nonconforming low-order finite element approximation for general quandrangular, hexahedral or simplicial meshes. For the energy balance equation, the scheme uses a discrete form of the conservation of the internal energy, which ensures that this latter variable remains positive; this relation includes a numerical corrective term, to allow the scheme to compute correct shock solutions in the Euler limit. The scheme is shown to have at least one solution, and to preserve the stability properties of the continuous problem, irrespectively of the space and time steps. In addition, it naturally boils down to a usual projection scheme in the limit of vanishing Mach numbers. Numerical tests confirm its potentialities, both in the viscous incompressible and Euler limits.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.9
Classification: 65M08,  76N15,  65M12,  76N19
Keywords: Compressible Navier-Stokes equations, pressure correction schemes, finite volumes, MAC scheme, finite elements.
@article{SMAI-JCM_2016__2__51_0,
author = {Dionysis Grapsas and Rapha\ele Herbin and Walid Kheriji and Jean-Claude Latch\'e},
title = {An unconditionally stable staggered pressure correction scheme for the compressible {Navier-Stokes} equations},
journal = {The SMAI journal of computational mathematics},
pages = {51--97},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {2},
year = {2016},
doi = {10.5802/smai-jcm.9},
mrnumber = {3633545},
zbl = {1416.76149},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.9/}
}
Dionysis Grapsas; Raphaèle Herbin; Walid Kheriji; Jean-Claude Latché. An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations. The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 51-97. doi : 10.5802/smai-jcm.9. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.9/`

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