# A polynomial spectral method for the spatially homogeneous Boltzmann equation

## TL;DR

This paper introduces a spectral Petrov-Galerkin method for the Boltzmann collision operator that achieves high accuracy and conservation properties by expanding the distribution function in orthogonal polynomials, with efficient algorithms for various collision kernels.

## Contribution

The paper develops a high-order spectral method using orthogonal polynomials for the Boltzmann equation, ensuring conservation and efficiency across different collision kernels.

## Key findings

- Achieves high accuracy even at low expansion orders.
- Ensures conservation of mass, momentum, and energy.
- Provides an algorithm with complexity O(N^7) and storage O(N^4).

## Abstract

We present a spectral Petrov-Galerkin method for the Boltzmann collision operator. We expand the density distribution $f$ to high order orthogonal polynomials multiplied by a Maxwellian. By that choice, we can approximate on the whole momentum domain $\mathbb{R}^3$ resulting in high accuracy at the evaluation of the collision operator. Additionally, the special choice of the test space naturally ensures conservation of mass, momentum and energy. By numerical examples we demonstrate the convergence (w.r.t. time) to the exact stationary solution. For efficiency we transfer between nodal and Maxwellian weighted Spherical Harmonics which are orthogonal w.r.t. the innermost integrals of the collision operator. Combined with efficient transformations between the bases and the calculation of the outer integrals this gives an algorithm of complexity $\mathcal{O}(N^7)$ and a storage requirement $\mathcal{O}(N^4)$ for the evaluation of the non linear Boltzmann collision operator. The presented method is applicable to a general class of collision kernels, among others including Maxwell molecules, hard and variable hard spheres molecules. Although faster methods are available, we obtain high accuracy even for very low expansion orders.