On the Use of Hermite Functions for the Vlasov–Poisson System

Abstract

We apply a second-order semi-Lagrangian spectral method for the Vlasov–Poisson system, by implementing Hermite functions in the approximation of the distribution function with respect to the velocity variable. Numerical tests are performed on a standard benchmark problem, namely the two-stream instability test case. The approach uses two independent sets of Hermite functions, based on Gaussian weights symmetrically placed with respect to the zero velocity level. An experimental analysis is conducted to detect a reasonable location of the two weights in order to improve the approximation properties.

Stability and conservation properties of Hermite-based approximations of the Vlasov-Poisson system

“…By isolating the effect of the first Fourier coefficients, we can prove Poincaré-type inequalities for a linear combination of Hermite polynomials and their first derivatives with respect to the norm induced by the weighted L 2 inner product where the weight is equal to e −v 2 . Indeed, the orthogonality of the first derivatives of the Hermite polynomials, equation (11), and the fact that 2n ≥ 2 for n ≥ 1, imply that:…”

“…where we noted that 2n + 1 ≤ 2n + n = 3n, since n ≥ 1. The last equality follows from (11). In short, we can write:…”

“…Empirical evidence that a convenient choice of the scaling factor α can improve the accuracy in Hermite discretizations of the Vlasov equation was shown in [35]. Generalized basis function of Hermite type has been investigated for solving time-dependent parabolic problems in [25] and, more recently, in [11] for the approximation of the Vlasov phase space. An adaptive strategy is currently under investigation, see [31], where both u and α may change through momentum and energy following how the plasma evolves in time during a numerical simulation.…”

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

“…This fact was exploited in successive works for investigating the properties of Vlasov-based plasma models [1,6] and developing computationally efficient numerical methods (see, e.g. : [10,8,9,13,26,4], and the code implementations described in [25,47]).…”