| The solutions of the kinetic equations describing certain systems of particles are known to have well defined asymptotic behaviors in the continuum limit, i.e., as the mean free path goes to zero. However, numerical methods that are consistent with the kinetic equations can become inconsistent in the same limit. In this thesis, we first construct a family of time stepping methods for the linear Boltzmann equation that remain consistent in the continuum limit, and we prove the correctness of the asymptotic behavior of the fully discrete solutions found using these time stepping methods. We then consider stochastic methods and related finite difference methods that are based on a convection-collision splitting of the kinetic equation. We show that the condensed history algorithm is inconsistent with the long time continuum limit of the linear Boltzmann equation unless the convection speed is modified as the mean free path tends to zero. We derive the speed correction for the condensed history algorithm and for several related finite difference methods. Finally, we apply one of these methods to the (nonlinear) planar Broadwell model of the nonlinear Boltzmann equation. We show that no speed correction is needed for this equation, and we compute the solution for small mean free path and for the continuum limit. |
