Convergence and error estimates for the Lagrangian based Conservative Spectral method for Boltzmann Equations

We develop error estimates for the semi-discrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by the authors in \cite{GT09}. In addition we study the long time convergence of such semi-discrete solution to equilibrium Maxwellian distribution that conserves the mass, momentum and energy associated to the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely conservation of mass, momentum and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semi-discrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani et al. in \cite[Chapter 12]{CIP} about finding error estimates for a numerical scheme associated to the Boltzmann equation, as well as showing the semi-discrete numerical solution converges to the equilibrium Maxwellian distribution associated to the initial value problem.