An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

D Grapsas (1, 2) , Raphaele Herbin (2) , W Kheriji (1) , J.-C Latché (1)

1 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
2 I2M - Institut de Mathématiques de Marseille

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 solution 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.

Mots clés: pressure correction schemes; Compressible Navier-Stokes equations; MAC scheme; finite elements; finite volumes

Domaines: Analyse numérique [math.NA]

D Grapsas, Raphaele Herbin, W Kheriji, J.-C Latché. An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations. SMAI Journal of Computational Mathematics, 2016, 2, pp.51-97. ⟨10.5802/smai-jcm.9⟩. ⟨hal-01115250⟩