A new method has been developed to incorporate particles with internal structure into the framework of the Variance Reduction method [17] for solving the discrete velocity Boltzmann Equation. Internal structure in the present context refers to physical phenomena like rotation and vibration of molecules consisting of two or more atoms. A gas in equilibrium has all modes of internal energy at the same temperature as the translational temperature. If the gas is in a non-equilibrium state, translational temperature and internal temperatures tend to proceed towards an equilibrium state during equilibration, but they all do so at different relaxation rates. In this thesis, rotational energy of a distribution of molecules is modeled as a single value at a point in a discrete velocity space; this represents the average rotational energy of molecules at that specific velocity. Inelastic collisions are the sole mechanism of translational and rotational energy exchange, and are governed by a modified Landau-Teller equation. The method is tested for heat bath simulations, or homogeneous relaxations, and one dimensional shock problems. Homogeneous relaxations demonstrate that the rotational and translational temperatures equilibrate to the correct final temperature, which can be predicted by conservation of energy. Moreover, the rates of relaxation agree with the direct simulation Monte Carlo (DSMC) method with internal energy for the same input parameters. Using a fourth order method for convecting mass along with its corresponding internal energy, a one dimensional Mach 1.71 normal shock is simulated. Once the translational and rotational temperatures equilibrate downstream, the temperature, density and velocity, predicted by the Rankine-Hugoniot conditions, are obtained to within an error of 0.5%. The result is compared to a normal shock with the same upstream flow properties generated by the DSMC method. Internal vibrational energy and a method to use Larsen Borgnakke statistical sampling for inelastic collisions is formulated in this text and prepared in the code, but remains to be tested. / text

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.

A new discrete velocity scheme for solving the Boltzmann equation has been implemented for homogeneous relaxation and one-dimensional problems. Directly solving the Boltzmann equation is computationally expensive because in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. To best solve the collision integral, collisions between each point in velocity space with all other points in velocity space must be considered, but this is very expensive. Motivated by the Direct Simulation Monte Carlo (DSMC) method, the computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space. A collision partner selection algorithm was implemented to favor collision partners that contribute more to the collision integral. The new scheme has a built in flexibility, where the resolution in approximating the collision integral can be adjusted by changing how many collision partners are sampled. The computational cost associated with evaluation of the collision integral is compared to the corresponding statistical error. Having a fixed set of velocities can artificially limit the collision outcomes by restricting post collision velocities to those that satisfy the conservation equations and lie precisely on the grid. A new velocity interpolation algorithm enables us to map velocities that do not lie on the grid to nearby grid points while preserving mass, momentum, and energy. This allows for arbitrary post-collision velocities that lie between grid points or completely outside of the velocity space to be projected back onto the nearby grid points. The present scheme is applied to homogeneous relaxation of the non-equilibrium Bobylev Krook-Wu distribution, and the numerical results agree well with the analytic solution. After verifying the proposed method for spatially homogeneous relaxation problems, the scheme was then used to solve a 1D traveling shock. The jump conditions across the shock match the Rankine-Hugoniot jump conditions. The internal shock wave structure was then compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 6. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Although using a relatively coarse approximation for the collision integral is computationally efficient, statistical noise pollutes the solution. The effects of using coarse and fine approximations for the collision integral are examined and it is found that by coarsely evaluating the collision integral, the computational time can be reduced by nearly two orders of magnitude while retaining relatively smooth macroscopic properties. / text

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.