### 2 Inria Tonus team

#### 2.1 Presentation

When I arrived in Strasbourg in 2006, Eric Sonnendrücker was head of an Inria project-team CALVI (CALcul scientifique et VIzualization). In 2012 he obtained a position in Garching at the Max-Planck-Institut für Plasmaphysik.

I became head of CALVI. See https://www.inria.fr/en/teams/calvi .

I submitted a new project after the final evaluation of CALVI in 2013. The new project TONUS (TOkamak NUmerical Simulation) was accepted in 2014. See https://www.inria.fr/en/teams/tonus.

As of December 2016, TONUS is composed of the following permanent researchers:

- Philippe Helluy, head;
- Matthieu Boileau https://www.researchgate.net/profile/Matthieu_Boileau;
- Emmanuel Franck http://www-irma.u-strasbg.fr/~franck/;
- Michaël Gutnic http://www-irma.u-strasbg.fr/php/home.php?qui=gutnic;
- Sever Hirstoaga http://www-irma.u-strasbg.fr/~hirstoag/;
- Michel Mehrenberger http://www-irma.u-strasbg.fr/~mehrenbe/;
- Laurent Navoret http://www-irma.u-strasbg.fr/~navoret/.

Post-doc researchers: David Coulette, Laura Mendoza.

PhD: Nicolas Bouzat, Ksander Ejjaaouani, Pierre Gerhard, Conrad Hillairet, Michel Massaro, Nhung Pham, Bruno Weber.

#### 2.2 Objectives

This project is related to the construction in France of the International Thermonuclear
Experimental
Reactor (ITER). This international project aims at producing thermonuclear fusion reactions in a hot
hydrogen plasma (temperature≃ 150 ×
10^{6}K). In the
long term it might become a way to produce clean
energy.

The plasma is confined with strong magnetic fields in a doughnut-shaped device: a tokamak. The main mathematical model for computing the plasma evolution is the Vlasov equation. Its unknown is the distribution function f(x,v,t) that counts the number of ions at point x and time t having velocity v. The problem is time-dependent in a six-dimensional phase space. The Vlasov equation reads

where B is the given magnetic field imposed by the tokamak superconducting coils, Φ is the electric potential, solution of the Poisson equation

and C(f) is a collision source term. This simple mathematical model leads to interesting mathematical problems: asymptotic limits for strong magnetic fields, large or small collision rates, etc.

Because it is set in a high-dimensional phase space, it is also a challenge for HPC. It requires the full power of the biggest supercomputers for obtaining realistic simulations.

#### 2.3 Achievements

In TONUS, we have obtained new results on the mathematical analysis of plasma models.

We have also proposed new numerical schemes for plasma physics.

CALVI and now TONUS are the advocates of semi-Lagrangian methods for solving kinetic equations. The methods are implemented in the Selalib library, which is a joint software project between Inria and the Max-Planck-Institut für Plasmaphysik in Garching. The most efficient semi-Lagrangian methods are generally transferred into GYSELA the production code of CEA (French atomic agency) for tokamak simulations.

For more details, we refer to the web pages of CALVI and TONUS and to the series of annual reports:

We also refer to the web page of Selalib http://selalib.gforge.inria.fr.

More recently, in order to handle more complex geometry, we have started to develop DG solvers for the Vlasov equations. The new developments are included in SCHNAPS the other main software project of the TONUS team: