On the velocity space discretization for the Vlasov-Poisson system: comparison between Hermite spectral and Particle-in-Cell methods. Part 2: fully-implicit scheme

We describe a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis, and the configuration space is discretized via a Fourier decomposition. The novelty of our approach is an implicit time discretization that allows exact conservation of charge, momentum and energy. The computational efficiency and the cost-effectiveness of this method are compared to the fully-implicit PIC method recently introduced by Markidis and Lapenta (2011) and Chen et al. (2011). The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability. The Fourier-Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to PIC. This paper concludes the study presented in Camporeale et al. (2013) where the same method has been described for a semi-implicit time discretization, and was compared against an explicit PIC.Comment: submitted to Journal of Computational Physics 16 pages, 7 figures. arXiv admin note: text overlap with arXiv:1311.209

On the velocity space discretization for the Vlasov-Poisson system: comparison between Hermite spectral and Particle-in-Cell methods. Part 1: semi-implicit scheme

We discuss a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis. We describe a semi-implicit time discretization that extends the range of numerical stability relative to an explicit scheme. We also introduce and discuss the effects of an artificial collisional operator, which is necessary to take care of the velocity space filamentation problem, unavoidable in collisionless plasmas. The computational efficiency and the cost-effectiveness of this method are compared to a Particle-in-Cell (PIC) method in the case of a two-dimensional phase space. The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability, and plasma echo. The Hermite spectral method can achieve solutions that are several orders of magnitude more accurate (at a fraction of the cost) with respect to the PIC method.Comment: 29 pages; 13 figures; submitted to Journal of Computational Physic