# A discontinuous Galerkin fast spectral method for the multi-species Boltzmann equation

## TL;DR

This paper presents a fast spectral method combined with discontinuous Galerkin discretization for solving the multi-species Boltzmann equation, achieving high accuracy and efficiency in simulating gas mixtures with various collision kernels.

## Contribution

It introduces a novel fast Fourier spectral method for the multi-species Boltzmann collision operator, coupled with DG discretization, improving computational efficiency and applicability to complex collision kernels.

## Key findings

- Achieves spectral accuracy and reduced computational complexity.
- Demonstrates efficiency and accuracy through numerical benchmarks.
- Validates the method against DSMC simulations across various scenarios.

## Abstract

We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the riveting properties of the single-species fast spectral method (Gamba et al. SIAM J. Sci. Comput., 39 pp. B658--B674 2017) including: (a) spectral accuracy, (b) reduced computational complexity compared to direct spectral method, (c) reduced memory requirement in the precomputation, and (d) applicability to general collision kernels. The fast collision algorithm is then coupled with discontinuous Galerkin discretization in the physical space (Jaiswal et al. J. Comp. Phys., 378 pp. 178--208 2019) to result in a highly accurate deterministic method (DGFS) for the full Boltzmann equation of gas mixtures. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method. Various benchmarks highlighting different collision kernels, different mass ratios, momentum transfer, heat transfer, and in particular the diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method.