# An entropic fourier method for the Boltzmann equation

## TL;DR

This paper introduces an entropic Fourier method for the Boltzmann equation that combines the advantages of discrete velocity and Fourier methods, ensuring positivity, entropy decay, and computational efficiency.

## Contribution

It develops a novel entropic Fourier approach that preserves positivity and the H-theorem while enabling fast Fourier transform algorithms for the Boltzmann collision operator.

## Key findings

- Preserves positivity of solutions
- Satisfies a discrete H-theorem
- Achieves second-order convergence rate

## Abstract

We propose an entropic Fourier method for the numerical discretization of the Boltzmann collision operator. The method, which is obtained by modifying a Fourier Galerkin method to match the form of the discrete velocity method, can be viewed both as a discrete velocity method and as a Fourier method. As a discrete velocity method, it preserves the positivity of the solution and satisfies a discrete version of the H-theorem. As a Fourier method, it allows one to readily apply the FFT-based fast algorithms. A second-order convergence rate is validated by numerical experiments