Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit

TL;DR

This paper develops a Fourier spectral method for the non cut-off Boltzmann equation that accurately transitions to the Fokker-Planck-Landau equation in the grazing collision limit, ensuring computational efficiency and uniform accuracy.

Contribution

It introduces a spectral method that correctly captures the grazing collision limit and provides a fast algorithm for kernel mode computation.

Findings

The spectral method converges to the Fokker-Planck-Landau equation in the grazing limit.

The method achieves uniform spectral accuracy across the grazing collision parameter.

A fast algorithm for kernel mode computation is derived.

Abstract

In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of L. Pareschi, B.Perthame, TTSP 25, (1996) and L.Pareschi, G.Russo, SINUM 37, (2000). We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved.