Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [F. Filbet and G. Russo, Kinet. Relat. Models, 2 (2009), pp. 231–250; G. Russo and P. Santagati, A new class of large time step methods for the BGK models of the Boltzmann equation, arXiv:1103.5247; P. Santagati, High Order Semi-Lagrangian Methods for the BGK Model of the Boltzmann Equation, Ph.D. thesis, University of Catania, Italy, 2007]. These methods work in a satisfactory way either in a rarefied or a fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper, we prove that the discrete solution of the scheme converges in a weighted $L^1$ norm to the unique smooth solution by deriving an explicit error estimate.

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