Semi-Lagrangian methods have proven useful for simulating the development of turbulence in magnetic fusion plasmas by solving the gyrokinetic Vlasov approximation on a mesh of the reduced phase space. The GYSELA code is based on this technique. Several variants of the classical split method by Cheng and Knorr for the Vlasov-Poisson equations have been introduced to retain good features of the solution in this setting. In particular fully conservative methods have been introduced. Because of the anisotropy of the solution and the preferred directions of motion of the particles in a tokamak, numerical errors can be reduced considerably for a given mesh size by adapting the mesh to the solution. We have in particular proposed a numerical scheme that can conserve exactly the equilibrium when a adequate mesh is used. We shall give an overview of the latest progress in these topics.
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