Nonlinear approximation theory for the homogeneous Boltzmann equation

A challenging problem in solving the Boltzmann equation numerically is that the velocity space is approximated by a finite region. Therefore, most methods are based on a truncation technique and the computational cost is then very high if the velocity domain is large. Moreover, sometimes, non-physical conditions have to be imposed on the equation in order to keep the velocity domain bounded. In this paper, we introduce the first nonlinear approximation theory for the Boltzmann equation. Our nonlinear wavelet approximation is non-truncated and based on a nonlinear, adaptive spectral method associated with a new wavelet filtering technique and a new formulation of the equation. A complete and new theory to study the method is provided. The method is proved to converge and perfectly preserve most of the properties of the homogeneous Boltzmann equation. It could also be considered as a general frame work for approximating kinetic integral equations.