Section: New Results

Preconditioning and implicit solvers

Participants : Emmanuel Franck [correspondent] , Philippe Helluy, Matthias Hoelzl, Ahmed Ratnani, Malcolm Roberts, Eric Sonnendrücker, Stefano Serra-Capizzano.

The Viscous-resistive MHD model used to simulate the instabilities is a multi-scale models with fast waves. In this context, it is not possible to use full explicit time schemes. However the classical implicit schemes are not usable directly since the matrices are ill-conditioned. For this reason it is necessary to use a preconditioning method. During this year we have studied a method called "physic based preconditioning" for the wave equations which consists to approximate the solution by suitable smaller and simpler systems. The results are very good. After this, we have extended this method to the Linearized Euler equation. During this new study, we have found additional difficulties which appear in some regimes. Two methods to treat this problem will be tested in 2016. We have also implemented a version of this preconditioning for the reduced MHD models of JOREK. The first results are positive. To finish, we have begun a collaboration with S. Serra-Capizzano to study at the theoretical level the physic based preconditioning and propose new preconditioning for each sub-systems of the Physic-Based PC efficient in all the physics regimes and for an arbitrary order.

We have also developed an implicit solver for the transport equation based on the upwind nature of the DG numerical flux. This solver will be used for solving Vlasov models or fluid models thanks to the Lattice-Boltzmann methodology. We have obtained recently a SPPEXA support (http://www.sppexa.de ) in a joint french-german-japanese project.