My research interests were mainly motivated by my background and the nature of the NMPP that should serve as a link between mathematicians (pure and applied), physicists and computer scientists. Working on the interface of so many fields require a lot of work and is known to be a complex task, but not a difficult one. Such collaboration can be done through scientific codes that we write together with our collaborators. By doing so, we create a bridge where physicists and mathematicians can focus on their work without taking care of complex requirements imposed by modern scientific applications and the different well known walls of super computers.

At the Max-Planck IPP, I am working on numerical discretizations of MHD and Maxwell equations. I am interested in hot plasmas such as the ones we find in Nuclear Fusion, which are motivated by the ITER project. The purpose is to build a power plant based on nuclear fusion in the context of Tokamaks. In fact, at sufficiently high energies (and temperature) Deuterium and Tritium can fuse to Helium, following the nuclear fusion reaction

At those energies, atoms are ionized and form what is known as a **plasma**.

In order to maintain particles inside a *chamber*, the Tokamak concept was invented, where particles are being trapped by a very strong magnetic field

Inside a Tokamak, particles follow *Banana* trajectories

Starting from Vlasov-Maxwell equations, one can derive different models depending on the interested physics. In my case, I am working on the MHD which can be viewed as the combination of Navier-Stokes and Maxwell equations. Studying such models is a complex task due to the different numerical challenges that appear in MHD. For instance, Explicit time schemes would lead to very small CFL numbers and would be CPU consuming. On the other hand, although implicit time schemes will avoid the CFL condition by filtering the fast phenomena, one still need to invert very large matrices, for which direct solvers are inappropriate due to the size of the reactor. These matrices are known to be very ill-conditioned because of the different scales and interplay between the different differential operators (transport, anisotropy, …). In addition, if one is interested in long time simulations, then having numerical schemes that conserve energy (a well defined one) is very important for the numerical stability. Such approaches are known as structure-preserving methods and are gaining a huge success these last years.

A typical numerical simulation in out context, is to start from an equilibrium then perturbed and study the grow of some given modes, which will lead to some energy exchange between different modes. Such studies are critical when studying the energy deposit on the divertors, in order to find better ways to confine the plasma.

More details can be found in the following sections, which describe the different work that I’ve been conducting since my Phd.