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

Abstract

Abstract. 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.Mathematics Subject Classification. 35Q30, 65N12, 65N30, 76M25.

Convergence of a mixed method for a semi-stationary compressible Stokes system

“…We will apply a topological degree argument, thereby reducing the proof to exhibiting a solution to a linear problem. The proof is similar to an argument employed in [16]. …”

“…The argument utilized in the second part of the above proof is similar to that found in [29]. Moreover, since the continuity scheme (3.1) is identical to a standard upwind finite volume discretization, the reader can consult [16] for a different approach to obtaining the positivity of the density. Remark 4.3.…”

“…where χ E is the characteristic function of E. The reader is referred to [16] for a recent paper using this formulation. We refer to [29] for higher order (polynomial) versions of the discontinuous Galerkin method.…”

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

“…We recall this result in the following theorem and refer to [10, chapter 5] for the general theory and [12,14] for its use in the case of other non-linear numerical scheme).…”

“…[17,19] and references herein. An extension to the barotropic NavierStokes equations close to the scheme developed here can be found in [14], together with references to related works. For stability reasons, the spatial discretization must preferably be based on pairs of velocity and pressure approximation spaces satisfying the so-called inf-sup or Babuska-Brezzi condition (e.g.…”