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.
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