A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

TL;DR

This paper develops a conservative spectral numerical scheme for the Boltzmann equation with anisotropic scattering, effectively modeling grazing collisions and converging to the Landau equation, with analysis of convergence rates and error comparisons.

Contribution

It introduces a new spectral method for anisotropic Boltzmann collisions that accurately approximates the Landau limit, including convergence analysis and error evaluation.

Findings

The scheme conserves mass, momentum, and energy.

The convergence rate depends on collision cross section parameters.

Rutherford scattering cross section has a logarithmic error in Landau approximation.

Abstract

We present the formulation of a conservative spectral scheme for Boltzmann collision operators with anisotropic scattering mechanisms to model grazing collision limit regimes approximating the solution to the Landau equation in space homogeneous setting. The scheme is based on the conservative spectral method of Gamba and Tharkabhushanam [17, 18]. This formulation is derived from the weak form of the Boltzmann equation, which can represent the collisional term as a weighted convolution in Fourier space. Within this framework, we also study the rate of convergence of the Fourier transform for the Boltzmann collision operator in the grazing collisions limit to the Fourier transform for the Landau collision operator for a family of non-integrable angular scattering cross sections. We analytically show that the decay rate to equilibrium depends on the parameters associated with the collision cross section, and specifically study numerically the differences between the classical Rutherford scattering angular cross section, which has logarithmic error in approximating Landau, and an artificial cross section with a linear error.