Vlasov Simulations Using Velocity-Scaled Hermite Representations

Abstract

The efficiency, accuracy, and stability of two different pseudo-spectral methods using scaled Hermite basis and weight functions, applied to the nonlinear Vlasov–Poisson equations in one dimension (1d-1v), are explored and compared. A variable velocity scaleUis introduced into the Hermite basis and is shown to yield orders of magnitude reduction in errors, as compared to linear kinetic theory, with no increase in workload. A set of Fourier–Hermite coefficients, representing a periodic Gaussian distribution function, are advanced through time with anO(Δt2)-accurate splitting method. Within this splitting scheme, the advection and acceleration terms of the Vlasov equation are solved separately using anO(Δt4)-accurate Runge–Kutta method. The asymmetrically weighted (AW) Hermite basis, which has been used previously by many authors, conserves particles and momentum exactly and total energy toO(Δt3); however, the AW Hermite method doesnotconserve the square integral of the distribution and is, in fact, numerically unstable. The symmetrically weighted (SW) Hermite algorithm, applied here to the Vlasov system for the first time, can either conserve particles and energy (forNueven) or momentum (forNuodd) as Δ t → 0, whereNuis the largest Hermite mode number. The SW Hermite method conserves the square integral of the distribution and, therefore, remains numerically stable. In addition, careful velocity scaling improves the conservation properties of the SW Hermite method. Damping and growth rates, oscillation frequencies, E-field saturation levels, and phase-space evolution are seen to be qualitatively correct during simulations. Relative errors with respect to linear Landau damping and linear bump-on-tail instability are shown to be less than 1% using only 64 velocity-scaled Hermite functions. Comparisons to particle-in-cell (PIC) simulations show that as the number of particles increases to more than 106, the PIC solutions converge to scaled SW Hermite solutions that were found in only 1/10 of the run-time. The SW Hermite method with velocity scaling is well-suited to kinetic simulations of warm plasmas.