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

Gallouët, Thierry; Gastaldo, Laura; Herbin, Raphaele; Latché, Jean-Claude

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 2, pp. 303-331.

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the $L^2$-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.

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