Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

“…We set up a semi-Lagrangian spectral-type method to find numerical approximations to the 1D-1V Vlasov-Poisson problem given by equations (1), (2), (3) and (4). The same algorithm was proposed in [25] within the framework of trigonometric functions, assuming a periodic distribution function f in both x and v variables. Instead, we will consider here a periodic function in x, and we will examine different sets of basis functions in the variable v and discuss pros and cons of the various approaches.…”

“…The extension to higher-dimensional problems (the 3D-3V case, for instance) is straightforward, though technically challenging in the implementation. The basic set up is discussed again in [25]. Imposing periodic boundary conditions for the variable x leads us to consider the domain: Ω x = [0, 2π[.…”

“…A semi-Lagrangian spectral method has been proposed in [25] for the numerical approximation of the nonrelativistic Vlasov-Poisson equations. These equations describe the dynamics of a collisionless plasma of charged particles, under the effect of a self-consistent electrostatic field [11].…”

“…For the exposition's sake, we assume that each plasma species (electrons and ions) is described by a 1D-1V distribution function, i.e., a function defined in a phase space that is one-dimensional in the independent variables x (space) and v (velocity). The approximation introduced in [25] has been specifically tested on Fourier-Fourier periodic discretizations for both variables in the phase space. The main goal of this paper is to extend our approach to Fourier-Legendre and Fourier-Hermite discretizations, i.e., by considering the spectral type representation with respect to v provided by Hermite and Legendre polynomials, in conjunction with a high-order semi-Lagrangian technique in time.…”

“…The method proposed in [25] works as follows. At each grid point, the updated value is set up to be equal to the value obtained by going backward, by a suitably small amount, along the local characteristic lines (approximated in a certain way).…”