Many aspects of the large-scale instabilities that appear in a magnetic confined plasma can be well described in the magneto-hydrodynamic (MHD) framework (with additional physics extensions). Solving these equations globally in the complicated geometry of a divertor tokamak or a stellerator is a highly demanding task due to the strong temporal and spatial multi-scale nature of the problem and the anisotropies reaching up to ten orders of magnitude which are introduced by the magnetic field. In general, explicit time integrators are not suitable while the use of implicit methods leads to very ill-conditioned matrices.
As a member (and leader) of the MHD group, I am working on different topics related to the discretization of MHD equations.
Numerical discretization of MHD equations
We are investigating different numerical schemes for the discretization of MHD equations and related Physics-Based preconditioning, splitting and relaxation schemes.
We are also studying the FEEC approach, and other discretizations like Powell-Sabin, Box-Splines.
We are developping new methods to construct (almost) automatically multi-patchs description of the magnetic flux using the Reeb graph and Morse theory.
Spectral Analysis of IGA discretizations
Since matrices associated to the discretization of MHD are ill-conditionned, I am also working on their spectral analysis, using the GLT method, in order to understand and design appropriate preconditioners.
We are developping different solvers on the top of a B-Splines geometric Multigrid.
Domain Specific Languages
Since modern architectures are getting more and more complicated, we are investigating and developping high level lanquages to handle this complexity in a transparent way, while allowing for symbolic computations of GLT symbols.