Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Abstract

A spectral method for the numerical solution of the multi-dimensional VlasovMaxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank-Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian-Free Newton-Krylov method.It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

Numerical Study of Inertial Kinetic-Alfvén Turbulence

“…The operator used in the present study is defined by Eq. 61 of (Delzanno 2015) and is constructed to conserve mass, energy, and momentum of each species. The collisionality parameter is ν/ω pe = 0.01.…”

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

“…On the other hand, spectral Transform methods use Hermite basis functions for unbounded domains, Legendre basis functions for bounded domains, and Fourier basis functions for periodic domains, and can outperform PIC in Vlasov-Poisson benchmarks [14,15]. Moreover, they can be extended in an almost straightforward way to multidimensional simulations of more complex models, like Vlasov-Maxwell [24]. Convergence of various formulations of these methods was shown in [31,46].…”

High‐Frequency Plasma Waves and Pitch Angle Scattering Induced by Pulsed Electron Beams

“…In order to verify these conclusions, we perform numerical simulations with the SpectralPlasmaSolver (SPS) Vlasov code (Delzanno, ; Roytershteyn & Delzanno, ; Vencels et al, ). SPS uses a spectral decomposition of the plasma distribution function in terms of Hermite polynomials in velocity space, a Fourier decomposition in physical space, and an implicit time discretization.…”

“…Note that in addition to the finite width, other physical effects that regularize the resonances at lh and uh , such as thermal effects, collisions, and nonlinear effects, may be important depending on specific conditions. In order to verify these conclusions, we perform numerical simulations with the SpectralPlasmaSolver (SPS) Vlasov code (Delzanno, 2015;Roytershteyn & Delzanno, 2018;Vencels et al, 2016). SPS uses a spectral decomposition of the plasma distribution function in terms of Hermite polynomials in velocity space, a Fourier decomposition in physical space, and an implicit time discretization.…”