Particle simulations of the Kac model of the Boltzmann equation

We consider a simplified mathematical model of the Boltzmann equation which was introduced by Kac. A quasi-Monte-Carlo particle simulation using (t, m, s)-nets is described. The particle movement is shown to be an evaluation of the volume of a subset of I4=[0,1)4. An error bound for quasi-Monte-Carlo approximation of the volume of a generalized quadrant set is derived, when a (t, m, s)-net is used. Convergence of the simulation is proved when the particles are reordered according to their velocity in every time step. Quasi-Monte-Carlo and Monte-Carlo particle simulations are compared in computational experiments. The results indicate that quasi-Monte-Carlo simulation outperforms standard Monte-Carlo approaches.