A fast Fourier spectral method for the linearized Boltzmann collision operator

We introduce a fast Fourier spectral method to compute linearized collision operators of the Boltzmann equation for variable hard-sphere gases. While the state-of-the-art method provides a computational cost O(MN^4 log N), with N being the number of modes in each direction and M being the number of quadrature points on a hemisphere, our method reduces the cost to O(N^4 log N), removing the factor M, which could be large in our numerical tests. The method is applied in a numerical solver for the steady-state Boltzmann equation with quadratic collision operators. Numerical experiments for both spatially homogeneous and inhomogeneous Boltzmann equations have been carried out to test the accuracy and efficiency of our method.