The M.A.C. scheme (called FDV in Trio_U) has been proposed by Harlow & Welch in 1965. This scheme is very robust and has been successfully generalized to many physical models (single-phase, two-phase, compressible, incompressible, turbulent, laminar, …).  The reasons of the high performances of this method are still mysterious.  Its main known flaws are that (i) it does not accurately treat flows at high Mach number (for which hyperbolic solvers are preferred) and (ii) it is restricted cartesian structured grids, which makes it difficult to use in complex geometries and with adaptive mesh refinement. The Immersed Boundary Methods, Fictitious Domain Methods, etc. have been developed with some success to overcome the issue of the treatment of complex geometries.

Within the Trio_U project, we are interested in identifying the properties of the M.A.C. scheme that makes so robust and to develop a numerical method that satisfies these properties and that can handle unstructured meshes. We are also interested in fictitious boundary methods, in particular to treat moving boundaries.

It is currently considered that the very reason of the robustness of the M.A.C. scheme comes from it ability to correctly decompose the space L²(W) (W is the fluid domain):

"¦Î L²(W) , $(j,y)Î (H¹(W))² : ¦=Ñj+Ñ´y

Indeed, this decomposition appears very often:

1. Moving reference frames (e.g. centrifugal force)
2. Surface tension (parasitic currents)
3. Taylor eddies (uÑu=-Ñp)
4. Equilibrium under gravity (Ñp=rg)

Our goal is thus to design a numerical that has the following properties:

1. High order decomposition of L²(W)
2. Stability (inf sup (BBL) condition)
3. Robustness (compressible, turbulent, single and two-phase flows)
4. Performance and parallel computing (Domain decomposition, Mortar)

The element developed is an improvement of the Crouzeix Raviart element. The velocity is P1NC and the pressure is P1+P0 in two dimensions and P1+P0+Pa in three dimensions.  This element has the required properties and can be interpreted, either in the framework of Galerkin Finite Element Methods, or in the framework of Finite Volume Methods (not staggered).  It is currently being tested in single-phase-flow configurations.

1. Two-fluid model
2. Wall functions
3. Mortar
4. ConvectionFormulation six équations