Hyperbolic Approximation of the Fourier Transformed Vlasov Equation

We construct an hyperbolic approximation of the Vlasov equation in which the dependency on the velocity variable is removed. The model is constructed from the Vlasov equation after a Fourier transformation in the velocity variable [9]. A well-chosen nite element semi-discretization in the spectral variable leads to an hyperbolic system.The resulting model enjoys 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: Landau damping, two-stream instability. Introduction Solving the Vlasov-Poisson equation is challenging. Some popular methods for studying this equation are the Particle-In-Cell (PIC) method [1] or the semi-lagrangian approach [5]. In a previous work [8], we constructed a reduced Vlasov-Poisson model with a velocity basis expansion. In this paper, we consider a Fourier velocity transformation of the Vlasov equation. We construct a reduced model where the unknown depends on space and time instead of the full phase-space variables. The reduced model is a linear hyperbolic system, with non-linear source terms. We present numerical results for classical plasma physics test cases. 1. Plasma mathematical model In our work, we consider the one-dimensional Vlasov equation ∂tf + v∂xf + E∂vf = 0, (1) where the unknown distribution function f depends on the space variable x ∈ R/LZ, the velocity variable v ∈ R and the time variable t ∈ R. The electric eld E depends on x and t and is the solution of the Poisson equation ∂xE = −1 + ˆ