Vlasov Simulations Using Velocity-Scaled Hermite Representations

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

“…We will not give further discussion on this issue due to the limitation of space, but just point out a recent paper of Schumer and Holloway [36] where a variable scaling factor for the Hermite basis was constructed for solving the nonlinear Vlasov-Poisson equations. The principal ideas in [36] are also useful for Hermite spectral approximations to the Fokker-Planck equations. Scaling factors are also used in a recent work of Shen [37] for the Laguerre spectral approximations, which also greatly enhance the resolution capacities of the Laguerre functions.…”

“…Spectral methods based on Hermite functions have been implemented before but were dismissed because of their poor resolution properties [19,18]. However, recent works of Tang [39] and Holloway et al [26,36] suggest that with proper selection of the scaling factors the Hermite basis can be quite competitive when modeling functions with Gaussian-shaped profiles. In solving the Vlasov equations [18,27], it was found that without careful velocity scaling of the Hermite functions, spectral expansions with 500 to 1500 Hermite modes are required to achieve only moderate accuracy levels.…”

“…In solving the Vlasov equations [18,27], it was found that without careful velocity scaling of the Hermite functions, spectral expansions with 500 to 1500 Hermite modes are required to achieve only moderate accuracy levels. For plasma kinetics simulation, a recent paper by Schumer and Holloway [36] indicates that the Hermite based spectral methods are very efficient and extremely stable when velocity scaling and symmetric weighting are used.…”

On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods

“…In Ref. [50] two different Hermite bases (so called 'asymmetrically' and 'symmetrically' weighted) and their properties were discussed, and a qualitative comparison between the Fourier-Hermite (FH) method and the PIC method was presented for simulations of Landau damping and bump-on-tail instability. Here we expand that work presenting an implicit time discretization, and quantitatively comparing the new scheme with an implicit PIC.…”

“…When comparing different schemes, the key information is represented by the CPU time needed to obtain a solution with a certain accuracy (this metric was not considered in [50]), in order to be able to assess which method must be preferred (and for which conditions). Hence, our comparisons between FH and PIC are presented in terms of computational efficiency and efficacy.…”