A fast spectral method for the inelastic Boltzmann collision operator and application to heated granular gases

In this paper, we propose a simple fast Fourier spectral method for the inelastic Boltzmann collision operator, with its application to one of the widely used models of granular gases, the inelastic Boltzmann equation with a heating source. Compared to the direct Fourier spectral method, our fast algorithm reduces the computational complexity from O (N⁶) to O (MN⁴ log ⁡ N) per evaluation of the collision operator in three dimensions, where N is the number of discretization points in each velocity dimension and M ≪N² is the number of quadrature points used on the unit sphere. We test the numerical accuracy and efficiency of the proposed method in both two dimensional and three dimensional examples, where in the latter case the famous Haff's cooling law for granular flows is successfully recovered.