Hyperbolic approximation of the fourier transformed vlasov equation

We construct an hyperbolic approximation of the Fourier transformed Vlasov equation in which the dependency on the spectral-velocity variable is removed. The model is constructed from the Vlasov equation after a Fourier transformation in the velocity variable as in [9]. A well-chosen finite element semi-discretization in the spectral variable leads to an hyperbolic system. The resulting model has interesting conservation and stability properties. It can be numerically solved by standard schemes for hyperbolic systems. We present numerical results for one-dimensional classical test cases in plasma physics: the Landau damping and the two-stream instability