Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation

We consider the nonlinear spatially homogeneous Boltzmann equation and its Fourier spectral discretization in velocity space involving periodic continuation of the density and a truncation of the collision operator. We allow discretization based on arbitrary sets of active Fourier modes with particular emphasis on the family of so-called hyperbolic cross approximations. We also discuss an offset method that takes advantage of the known equilibrium solutions. Extending the analysis in Filbet & Mouhot (2011, Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc., 363, 1947-1980), we establish consistency estimates for the discrete collision operators and stability of the semidiscrete evolution. Under an assumption of Gaussian-like decay of the discrete solution, we give a detailed bound for Hs-Sobolev norms of the error due to Fourier spectral discretization.