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The Boltzmann equation on a two-dimensional lattice; theoretical and numerical results

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. This paper contains a brief description of the proof that was recently presented elsewhere ([L. Fainsilber, P. Kurlberg, B. Wennberg, preprint 2004]). It also presents the results of numerical experiments.