SUMMARYWe develop in this paper a discretization for the convection term in variable density unstationary Navier-Stokes equations, which applies to low-order non-conforming finite element approximations (the so-called Crouzeix-Raviart or Rannacher-Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L 2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L 2 norm, second-order space convergence for the velocity and first-order space convergence for the pressure are observed.

SUMMARY We propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non‐conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second‐order spatial convergence rate for the Laplace equation on any unstructured two‐dimensional/three‐dimensional grids. As in the MUSCL method, the numerical convective fluxes are built with a prediction‐limitation process, which ensures that the discrete maximum principle is satisfied for pure convection problems. The limitation does not involve any geometrical reconstruction, thus allowing the use of completely general grids, in any space dimension. Copyright © 2012 John Wiley & Sons, Ltd.

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International audienceWe propose a discretization for the MAC scheme of the viscous dissipation term τ (u) : ∇u (where τ (u) stands for the shear stress tensor associated to the velocity field u), which is suitable for the approximation of this term in a conservation equation for a scalar variable. This discretization enjoys the property that the integral over the computational domain Ω of the (discrete) dissipation term is equal to what is obtained when taking the inner product of the (discrete) momentum balance equation by u and integrating over Ω. As a consequence, it may be used as an ingredient to obtain an unconditionally stable scheme for the compressible Navier-Stokes equations. It is also shown, in some model cases, to ensure the strong convergence in L1 of the dissipation term