| The standard technique for the solution of kinetic theory and transport problems usually involves the expansion of the velocity distribution function (VDF) in some suitable set of basis functions.; The main objective of this thesis is to develop accurate and efficient numerical methods for the Boltzmann equation (BE) with the aim to calculate the distribution function.; For an ensemble of ions dilutely dispersed in a background of neutrals under the influence of an applied electric field, the exact solution of the Bhatnagar, Gross, and Krook (BGK) collision model is used to study different numerical solutions of the BE. In place of the traditional expansion techniques, a discretization of the VDF is proposed. The discretization is based upon a set of polynomial basis functions that yield the optimum convergence of the solution.; For electrons dilutely dispersed in a background of neutrals, the BE is approximated by a Fokker-Planck equation (FPE). A detailed comparison of the eigenvalues and eigenfunctions of the FP operator is carried out with different numerical methods including a finite difference (FD) scheme, Lagrange interpolation (LI) method, the Quadrature Discretization Method (QDM) with speed points and the QDM with Davydov points. The time dependent solutions of the FP are obtained with the different energy discretizations applied in conjunction with an appropriate time stepping procedure. The QDM with Davydov points provides a more rapid convergence of the eigenvalues and eigenfunctions relative to the other methods used.; The relaxation of a nonequilibrium distribution of electrons in a mixture of CCl{dollar}sb4{dollar} with either Ar or Ne is studied. The electron collisions in this analysis include elastic collisions between electrons and CCl{dollar}sb4{dollar} and the inert gas, vibrationally inelastic collisions between electrons and CCl{dollar}sb4,{dollar} as well as the electron attachment reaction with CCl{dollar}sb4.{dollar} The time dependent electron energy distribution function is determined from the BE and the energy relaxation times are determined.; In addition, the nature of the spectrum of the Boltzmann collision operator for realistic ion-atom interactions is studied with a discretization of the integral collision operator. The singular nature of the collision operator is considered in detail with a novel interpolation technique. The eigenvalues and eigenfunctions for the Henyey-Greenstein (HG) model differential cross section are calculated. The relaxation of anisotropic ion distributions with and without an applied electric field is studied. (Abstract shortened by UMI.)... |
