Fast Spectral Methods for the Fokker–Planck–Landau Collision Operator

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

“…Actually the Fourier transform of A(v) and f (v) are obtained as a by-product during the computation of Q( f ), if one applies a spectral scheme such as in [26]. Then the eigenvalue can be computed easily.…”

“…However this is not true for many efficient schemes. The spectral scheme on nFPL operator introduced in [26] preserves the mass while conservation of momentum and energy are "spectrally preserved". As for the FP operator, the discretization we are using (see section 3.2) preserves the mass while the errors in conservation of momentum and energy are O(∆v 2 ).…”

“…We use the fast spectral method designed by Pareschi, Russo and Toscani [26]. The computational cost is O (N log N), where N = N d v is the grid points in velocity space.…”

“…The spectral scheme described in [26] allows us to compute the nFPL operator (1.2) efficiently. Numerical experiments shows that N v = 32 can give satisfactory results.…”

“…In [31], [2], [10], [4], [5], [6], conservative finite difference type discretizations were applied to the space homogeneous equation was derived. To reduce the computational cost caused by the high dimensional integral in the collision operator, spectral schemes were derived in [25], [26], [15]. However all these (explicit) schemes suffer from the stability constraint ∆t < Cε∆v 2 .…”