Accurate numerical methods for the collisional motion of (heated) granular flows

In this paper, we extend the spectral method developed in [L. Pareschi, B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. Stat. Phys. 25 (1996) 369-383; L. Pareschi, G. Russo, Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal. 37 (2000) 1217-1245] to the case of the inelastic Boltzmann equation describing the collisional motion of a granular gas with and without a heating source. The schemes are based on a Fourier representation of the equation in the velocity space and provide a very accurate description of the time evolution of the distribution function. Several numerical results in dimension one to three show the efficiency and accuracy of the proposed algorithms. Some mathematical and physical conjectures are also addressed with the aid of the numerical simulations.