Optimized 1d-1v Vlasov-Poisson simulations using Fourier-Hermite spectral discretizations.

Abstract

A 1d-1v spatially-periodic, Maxwellian-like, charged particle phase-space distribution f(x, v, t) is represented by one of two different Fourier-Hermite basis sets (asymmetric or symmetric Hermite normalization) and evolved with a similarly transformed and filtered Vlasov-Poisson set of equations. The set of coefficients $f\sbsp{\alpha}{mn}(t)$ are advanced through time with an $O(\Delta t\sp2$)-accurate splitting method,$\sp1$ using a $O(\Delta t\sp4)$ Runge-Kutta time advancement scheme on the $v\partial\sb{x}f$ and $E\partial\sb{v}f$ terms separately, between which the self-consistent electric field is calculated. This method improves upon that of previous works by the combined use of two optimization techniques: exact Gaussian filtering$\sp2$ and variable velocity-scaled$\sp3$ Hermite basis functions.$\sp4$ The filter width, $v\sb{o}$, reduces the error introduced by the finite computational system, yet does not alter the low-order velocity modes; therefore, the self-consistent fields are not affected by the filtering. In addition, a variable velocity scale length U is introduced into the Hermite basis functions to provide improved spectral accuracy, yielding orders of magnitude reduction in the $L\sb2$-norm error.$\sp5$ The asymmetric Hermite algorithm conserves particles and momentum exactly, and total energy in the limit of continuous time. However, this method does not conserve the Casimir ${\int\int} f\sp2dxdu$, and is, in fact, numerically unstable. The symmetric Hermite algorithm can either conserve particles and energy or momentum (in the limit of continuous time), depending on the parity of the highest-order Hermite function. Its conservation properties improve greatly with the use of velocity filtering. Also, the symmetric Hermite method conserves ${\int\int} f\sp2dxdu$ and, therefore, remains numerically stable. Relative errors with respect to linear Landau damping and linear bump-on-tail instability are shown to be less than 1% (orders of magnitude lower than those found in comparable Fourier-Fourier and PIC schemes). Varying the Hermite velocity-scale and increasing the filtering can enhance accuracy and retain longer recursion times in the Landau damping cases. Saturation levels of the electric field and BGK evolution is seen to be qualitatively correct. ftn$\sp1$Cheng and Knorr, J Comp Phys 22 (1976). $\sp2$Klimas, J Comp Phys 68 (1987). $\sp3$Boyd, J Comp Phys 54 (1984). $\sp4$Armstrong and Montgomery, Phys Fluids 12 (1969). $\sp5$Holloway, Transp Theory Stat Phys (1996), Tang, SIAM J. of Scientific Comp. 14 (1993).