Conservative Deterministic Spectral Boltzmann Solver Near the Grazing Collisions Limit

TL;DR

This paper extends a spectral Boltzmann solver to handle anisotropic collisions and demonstrates its consistency with Fokker-Planck-Landau equations in the grazing collisions limit, improving accuracy near this regime.

Contribution

It introduces a generalized formulation for anisotropic scattering and derives the grazing collisions limit, linking the Boltzmann and Landau equations within a spectral method framework.

Findings

Extended the spectral method to anisotropic collisions.

Derived the grazing collisions limit for the spectral method.

Confirmed consistency with Fokker-Planck-Landau equations.

Abstract

We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. Within this framework we have extended the formulation to the case of more general case of collision operators with anisotropic scattering mechanisms, which requires a new formulation of the convolution weights. We also derive the grazing collisions limit for the method, and show that it is consistent with the Fokker-Planck-Landau equations as the grazing collisions parameter goes to zero.