Section: New Results

Numerical methods

Finite volume methods in curvilinear coordinates

Participants : Hervé Guillard, Boniface Nkonga, Afeintou Sangam.

Finite volume methods are specialized numerical techniques for the solving of divergence equations in strong conservation forms of the form

where $S$ is a scalar or a vector while $T$ is a vector or a second-order tensor. Using textbook formulas for the expression of the divergence operator in curvilinear coordinates, the use of these coordinate systems instead of the Cartesian one can lead to a loss of the strong conservation form of the equations and introduce a source term in (1 ). Actually, this is unnecessary and one can show that whatever the system of curvilinear coordinate used, there exists a strong conservation law form of the system. However, when vector equations have to be considered (that is if $S$ is a vector and $T$ a tensor), it is necessary to extract from (1 ) scalar equations for the components of the vector $S$ and this may destroy the strong conservation form of the equation. Following the work done in [12] where a general method (i.e that does not depend on the curvilinear system used) based on the projection of the discretized vector system have been designed, we have studied this year its application to cylindrical coordinates in the case where the geometry is a torus. This approach is robust and accurate for problems that take place for instance inside tokamak devices for magnetic confinement fusion or in toroidal plasmas occuring in stars and galaxies to take another examples. The method is now implemented in the PlaTo software.

Entropy preserving schemes for conservation laws

Participants : Christophe Berthon [University of Nantes] , Bruno Dubroca [CEA/DAM/CESTA and University of Bordeaux 1] , Afeintou Sangam.

Entropy preserving schemes for conservation laws

In collaboration with C. Berthon of University of Nantes, and B. Dubroca of CEA/DAM/CESTA and University of Bordeaux 1, we have established a new technique that proves discrete entropy inequalities of finite volume methods to approximate conservation laws. This technique is free of additional numerical models such as kinetic and relaxation schemes. Moreover, our results leads to a full class of entropy preserving schemes for general Euler equations [11] . This proposed thechnique has been successfully applied to two intermediates states scheme for 10-moments equations with laser source-term in context of Inertial Fusion Confinement. Moreover, the derived procedure is now extended to Saint-Venant model.

Mesh adaptation Methods

Participants : Anca Belme [Projet Tropics] , Hubert Alcin [Projet Tropics] , Alain Dervieux, Frédéric Alauzet [Projet Gamma, INRIA-Rocquencourt] .

This activity results from a cooperation between Gamma, Tropics, Pumas, and Lemma company. See details in Tropics and Gamma activity reports. Its concerns Pumas's subject through the current applications of mesh adaptation to flows with interfaces and the starting application of mesh adaptation to Large Eddy Simulation. It is also planned to use mesh adaptation for simplified plasma models in the context of ANEMOS ANR project.

Parallel CFD algorithms

Participants : Hubert Alcin [Tropics] , Olivier Allain [Lemma] , Anca Belme [Tropics] , Marianna Braza [IMF-Toulouse] , Alexandre Carabias [Tropics] , Alain Dervieux, Bruno Koobus [Université Montpellier 2] , Carine Moussaed [Université Montpellier 2] , Hilde Ouvrard [IMF-Toulouse] , Stephen Wornom [Lemma] .

Pumas is associated to the ANR ECINADS project started in end of 2009, devoted to the design of new solution algorithms for unsteady compressible flows, adapted to scalable parallelism and to reverse (adjoint) Automatic Differentiation. See in the activity report of Tropics. The newer two-level deflation algorithm is currently applied to a simplified plasma model in the context of ANEMOS ANR.