In the study of kinetic theory and especially in the study of rarefied gas dynamics one often turns to the Boltzmann equation. The mathematical theory developed by Ludwig Boltzmann was at first sight applicable in aerospace engineering and fluid mechanics. As of today, the methods in kinetic theory are extended to other fields, for instance, molecular biology and socioeconomics, which makes the need of finding efficient solution methods still important. In this thesis, we study the underlying theory of the continuous and discrete Boltzmann equation for monatomic gases. We extend the theory where needed, such that, we cover the case of colliding molecules that possess different levels of internal energy. Mainly, we discuss discrete velocity models and present explicit calculations for a model of a gas consisting of polyatomic molecules modelled with two levels of internal energy. / I studiet av kinetisk teori och speciellt i studiet av dynamik för tunna gaser vänder man sig ofta till Boltzmannekvationen. Den matematiska teorien utvecklad av Ludwig Boltzmann var vid första anblicken tillämpbar i flyg- och rymdteknik och strömningsmekanik. Idag generaliseras metoder i kinetisk teori till andra områden, till exempel inom molekylärbiologi och socioekonomi, vilket gör att vi har ett fortsatt behov av att finna effektiva lösningsmetoder. Vi studerar i denna uppsats den underliggande teorin av den kontinuerliga och diskreta Boltzmannekvationen för monatomiska gaser. Vi utvidgar teorin där det behövs för att täcka fallet då kolliderande molekyler innehar olika nivåer av intern energi. Vi diskuterar huvudsakligen diskreta hastighetsmodeller och presenterar explicita beräkningar för en modell av en gas bestående av polyatomiska molekyler modellerad med två lägen av intern energi.

We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models. Existence of solutions of weakly non-linear half-space problems for general DVMs are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found. Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution. The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.

In this thesis, we study the relationship between the generalized eigenvalue problem (GEP) $Ax=\lambda Bx$, and systems of differential equations. We examine both the Jordan canonical form and Kronecker's canonical form (KCF). The first part of this work provides an introduction to the fundamentals of generalized eigenvalue problems and methods for solving this problem. We discuss the QZ algorithm, which can be used to determine the generalized eigenvalues and also how it can be implemented on MATLAB with the built in function 'eig'.  One essential facet of this work is the exploration of symmetric matrix pencils, which arise when A and B are both symmetric matrices. Furthermore we discuss discrete velocity models (DVMs) focusing specifically on a 12-velocity model on the plane. The results obtained are then applied to half-space problems for discrete velocity models, with a focus on planar stationary systems.