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

Ricardo J. Alonso, Irene M. Gamba, Sri Harsha Tharkabhushanam

Published 2016-11-13Version 1

In this paper we study the approximation properties of the spectral conservative method for the elastic and inelastic Boltzmann problem introduced by the authors in \cite{GT09}. The method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction used for conservation of mass, momentum and energy. We present an analysis on the accuracy and consistency of the method, for both elastic and inelastic collisions, and a discussion of the $L^{1,2}$ theory for the scheme in the elastic case which includes the estimation of the negative mass generated by the scheme. This analysis allows us to present Sobolev convergence and error estimates for the numerical approximation. The estimates are based on recent progress of convolution and gain of integrability estimates by some of the authors and a corresponding moment inequality for the discretized collision operator. The Lagrangian optimization correction algorithm is not only crucial for the error estimates and the convergence to the equilibrium Maxwellian, but also it is absolutely necessary for the moments conservation for systems of kinetic equations in mixtures and chemical reactions. 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 Boltzmann scheme.