A fast spectral particle method for the Landau equation

Abstract

We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is $\mathcal{O}(N+M^d\log M)$, where $N$ is the number of particles and $M$ the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities. Two-dimensional numerical experiments with Maxwellian interactions illustrate the accuracy and efficiency of the method, its sensitivity to particle and spectral resolution, and the effects of filtering. Comparisons with a direct blob implementation demonstrate improved accuracy at a strongly reduced computational cost.