# Approximation to Singular Quadratic Collision Model in Fokker-Planck-Landau Equation

## TL;DR

This paper introduces a Hermite-Galerkin spectral method for efficiently solving the Fokker-Planck-Landau equation with a singular quadratic collision model, accurately capturing moments despite singularities.

## Contribution

It presents a novel spectral approach combining Burnett polynomials and Hermite expansion to handle singular collision models in kinetic equations.

## Key findings

- Accurately captures low-order moments.
- Handles very singular collision models smoothly.
- Demonstrates satisfactory accuracy and performance.

## Abstract

We propose a Hermite-Galerkin spectral method to numerically solve the spatially homogeneous Fokker-Planck-Landau equation with singular quadratic collision model. To compute the collision model, we adopt a novel approximation formulated by a combination of a simple linear term and a quadratic term very expensive to evaluate. Using the Hermite expansion, the quadratic term is evaluated exactly by calculating the spectral coefficients. To deal with singularities, we make use of Burnett polynomials so that even very singular collision model can be handled smoothly. Numerical examples demonstrate that our method can capture low-order moments with satisfactory accuracy and performance.