## Section: New Results

Keywords : Compact distribution schemes, multimaterial and multiphase flows (multimaterial flows, multiphase flows), mesh adaptation.

### Numerical schemes and algorithms for fluid mechanics.

Participants : Rémi Abgrall, Christophe Berthon, Andreas Bollerman, Benjamin Braconnier, Cecile Dobrzynski, Adam Larat, Mikaël Lutz, Boniface Nkonga, Mikaël Papin, François Pellegrini, Vincent Perrier, Pierre Ramet, Mario Ricchiuto, Jean Roman, Cédric Tavé.

#### Residual distribution schemes

This year many developments have been conducted and implemented in the
`FluidBox` software after [1] which has open up many
doors.

One may list the developments of stabilised and quasi monotone centered RD schemes [57] , [56] , the approximation of viscous terms that is consistent with what is done on the convective/hyperbolic part. This item has been worked out in collaboration with N. Villedieu and H. Deconinck from VKI and Jiri Dobes from CTU in Prag[71] , [59] . These schemes have all been extended to unsteady problems and quadrilateral meshes. Thanks to a careful analysis of the implicit phase of the scheme, we have been able to reduce the CPU cost by a half.

We have also started to work on very high order residual distribution schemes. Lagrange interpolant of degree 3 an 4 are considered with either an upwind or a centered formulation [37] , [38] .

The work on shallow water have been prolongated. The scheme is now able to handle dry beds. We have shown that genuinely steady 2D are numericaly preserved by the scheme [9] .

With C.W. Shu (Brown University), we have also developped a new method which is somewhat between the Residual distribution schemes that need continuous interpolant and the Discontinuous Galerkin ones where the interpolant is discontinuous (article in preparation).This method is currently second order in space.

Mario Ricchiuto has written in collaboration with H. Deconinck a chapter for the Encyclopedia of Computational Mechanics, [68] .

#### Relaxation schemes

The stability properties of this schemes, as well as the easyness with which they can be implemented makes them mandatory for many complex applications. This year two have been considered : multifluid flow problems [28] , [24] , and radiative transfert with CELIA [21] , [25] , [45] .

#### MUSCL type scheme

We have developped a numerical procedure for which we can prove that mass and pressure remain positive under a standard CFL type property. The method has been validated against many standard test cases [4] , [23] , [46] .

#### Interface and multifluid problems

In collaboration with CEA–CESTA, we are developping an interface tracking method using the level set method with the Ghost fluid technique. The method has been implemented in 2D and validated against several benchmark tests, in particular for interface between compressible and incompressible fluids[47] , [72] .

B. Braconnier is finishing his thesis on the development of numerical
procedures for the simulation of low mach number flows[55] . The method use
relaxation solvers. The scheme, which is implicit, uses the `PaStiX` library, and this strategy seems to be the most efficient. In the near
future, we expect to be able to consider larger problems by using the
iterative method library that is developped in the team.

#### Phase transition

V. Perrier has been studying the structure of discontinuities in the seven equation model and the five equation model using traveling wave techniques. This topic is done in collaboration with H. Guillard from the Smash project in Sophia Antipolis.

`PaStiX` and `FluidBox`

M. Papin has integrating the `PaStiX` high performance solver in the
`FluidBox` code.
The results have been presented in [48] . This technique will be used
within the ADIGMA project, as well as more recent work on iterative
solvers by J. Gaidamour, P. Hénon, P. Ramet and J. Roman.

#### Parallelization and fluid mechanics

We have developped a parallel strategy by domain decomposition which is tuned for aerodynamics problems (THOT code of CEA/CESTA). The problem was to construct domain decomposition criteria which preserve the multi-block structure of the mesh. The code is fully operational and will be delivered to CEA/CESTA in the near future.

#### Mesh adaptation

C. Dobrzynski has worked on fully parallel mesh adaptation procedure that uses standard sequential mesh adaptation codes. The idea is to adapt the mesh on each processor without changing the interfaces. Then the interfaces are modified. The main advantage is the simplicity because there is no need to parallelize mesh generation tools (insert/delete, swap, etc). The main techniques are described in [31] , [30] .

We also have started to work on the definition of an anisotropic metric which is computed from the he output of a Residual distribution code. Once this will be done, standard mesh adaptation method will be used so that the numerical error of the solution is controled.