A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

Abstract

We present the design and implementation of an L 2 -stable spectral method for the discretization of the Vlasov-Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L 2 -stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.

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

“…where w m = 2π/M in the periodic Fourier case (recall the quadrature formula (19)), while the weights are related to the quadrature formula (45), which are also valid for both Legendre and Hermite cases. From (35), at time step t k+1 we find out that: In fact, one has:…”

“…Within this approach, exact conservation laws in the discrete setting, i.e., discrete invariants in time for number of particles (mass, charge), momentum and energy, can be constructed from Hermite expansion's coefficients. Moreover, as pointed out in [57,56,24,45,46], expansions in Hermite basis functions are intrinsically multiscale, providing a natural connection between low-order moments of the plasma distribution function and typical fluid moments.…”

“…Semi-Lagrangian methods have been proposed in different frameworks such as the Finite Volume method [26,4], Discontinuous Galerkin method [2,3,39], finite difference methods based on ENO and WENO polynomial reconstructions [21], as well as in the propagation of solutions along the characteristics in an operator splitting context [1,17,20,28,27,52,23]. These methods offer an alternative to Particle-in-Cell (PIC) methods [22,58,59,12,18,19,42,43,47,53] and to the so called Transform methods based on spectral approximations [45,48,41,57,56]. PIC methods are very popular in the plasma physics community and are the most widely used methods because of their robustness and relative simplicity [6].…”

“…It should also be noted that Hermite-based method may suffer of instability issues. In alternative, the use of Legendre polynomials can offer stable spectral approximations, with almost the same good conservation properties of Hermite based discretizations (see [45,46]).…”

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

“…Based on the seminal paper [30], an alternative approach, called the Transform method, was developed at the end of the '60s, which uses a spectral decomposition of the distribution function and leads to a truncated set of moment equations for the expansion coefficients [2]. To this end, Hermite basis functions are used for unbounded domains, Legendre basis functions for bounded domains, and Fourier basis functions for periodic domains, see, e.g., [36,40,33,48,47]. These techniques can outperform PIC [13,14] in Vlasov-Poisson simulations.…”