New approaches to stable models for computational plasma physics

Due to the presence of multiple physical scales and complex nonlinear interactions, the numerical simulation of fusion plasmas often leads to computational problems of huge complexity. A long-standing challenge is then to design numerical methods that are computationally efficient, high order accurate and stable on very long time scales. Fortunately, steady progresses in the theory of structure-preserving discretizations have provided a solid mathematical ground for the development of stable high order numerical schemes. In this lecture I will give a brief review of the compatible Finite Element methods that have been developed in this direction, and I will explain how these tools are now being extended to design stable numerical models for the Vlasov-Maxwell equations. Recent ideas that allow to further improve the computational efficiency of such methods will be presented, along with a novel approach to low-noise particle approximations.