## Section: New Results

### Multiple time scales solvers and Magneto HydroDynamics

Participants : Jean-Philippe Braeunig, Emmanuel Frénod, Michaël Gutnic, Philippe Helluy, Alexandre Mouton, Eric Sonnendrücker.

#### Multiple time scales solvers

One of the most important difficulties of numerical simulation of magnetized plasmas is the existence of multiple time and space scales, which can be very different. In order to produce good simulations of these multiscale phenomena, we have investigated the development of models and numerical methods which are adapted to these problems. The two-scale convergence theory introduced by G. Nguetseng and G. Allaire is one of the tools which can be used to rigorously derive multiscale limits and to obtain new limit models which can be discretized with a usual numerical method: we call this procedure a two-scale numerical method. Within the thesis of Alexandre Mouton we developed a two-scale semi-Lagrangian method and applied it on a gyrokinetic Vlasov-like model in order to simulate a plasma submitted to a large external magnetic field. In order to tackle this complex model, we first investigated this idea in simpler cases. First, we developed a two-scale finite volume method applied to the weakly compressible 1D isentropic Euler equations. Even if this mathematical context is far from a Vlasov-like model, it is a relatively simple framework in order to study the behaviour of a two-scale numerical method in front of a nonlinear model. In a second part, we developed a two-scale semi-Lagrangian method for the two-scale model developed by E. Frénod, F. Salvarani et E. Sonnendrücker in order to simulate axisymmetric charged particle beams. Even if the studied physical phenomena are quite different from magnetic fusion experiments, the mathematical context of the one-dimensional paraxial Vlasov-Poisson model is very simple for establishing the basis of a two-scale semi-Lagrangian method. Finally, we used the two-scale convergence theory in order to improve M. Bostan's weak-* convergence results about the finite Larmor radius model, and we developed a forward semi-Lagrangian method implementing the two-scale method.

#### Magneto HydroDynamics

The MagnetoHydroDynamics (MHD) equations are a simplified but rich model of conducting fluids. They can be used in some parts of ITER but also in stellar physics, geophysics and plasmas physics. The Discontinuous Galerkin (DG) method is already used for solving MHD problems. It has proved to be very efficient and accurate. However, several difficulties are still present:

-in some cases, the MHD first order equations admit several entropy solutions. The DG method can be designed in order to satisfy an entropy principle, but it is not clear how it behaves in case of multiple solutions;

-if in some parts of the mesh small cells are required, it is important, for efficiency reasons to be able to deal with several time step sizes;

-finally, the divergence free condition is treated by the hyperbolic divergence cleaning technique (see references in [14] and [45] ).

As a conclusion, this project permitted to address relevant problems in the approximation of the MHD equations by DG schemes. The scheme has been implemented in CM2, a parallel and general purpose CFD code.

#### Numerical simulation of compressible multi-material fluid flows

**Participant:** Jean-Philippe Braeunig

This work is achieved in collaboration with J.-M. Ghidaglia (ENS Cachan), F. Dias (ENS cachan) and B. Desjardins (ENS Ulm) in the frame of the Laboratoire de Recherche Commun MESO (ENS cachan - CEA Bruyères-le-Châtel).

This collaboration has begun during Braeunig's PhD (2004-2007) which was a collaboration CEA-ENS Cachan. We have designed a novel pure eulerian Finite Volumes method for compressible multi-material fluid flows with sharp interface capturing called FVCF-NIP, (Finite Volumes with Characteristic Flux (VFFC) scheme of Ghidaglia et al, interface capturing Natural Interface Positioning (NIP)). A new concept is introduced, the condensate, which allows to handle mixed cells containing two or more materials and to calculate the evolution of the interface on the fixed eulerian grid. The main features of this method are: second order in time and space, local conservation of mass, momentum and total energy, no diffusion of any materials eulerian quantity on each others through the interface and free sliding of materials on each others at the interface. This work is described in Braeunig's PhD report 2007 and in a paper by Braeunig et al [24] published in 2009. A later INRIA report is published in 2009 by Braeunig [46] , that describes some improvements of the NIP interface capturing method. The method is currently improved by the work of trainees and post-docs at LRC MESO. The PhD thesis of Daniel Chauveheid has begun in September 2009 on this topic at CEA Bruyères-le-Châtel with the collaboration of Braeunig.

This method is used for industrial applications. One of them is the simulation of Liquefied Natural Gas (LNG) sloshing in LNG carriers (LNG tanks in a boat). The issue was to study and design a procedure to extrapolate experimental results obtained in laboratory with a small scale model (scale 1/40) to the scale of the LNG carrier (scale 1) taking into account compressible effects. A scaling law has been proposed and validated numerically with a VFFC-NIP code. The code has been validated by comparing results with analytical solutions and with a reduced numerical model on academic benchmarks. This work has been presented by J.-M. Ghidaglia at ISOPE 2010 Conference and described in a paper in the ISOPE 2010 Conference Proceeding, see Braeunig et al [38] .