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We consider a multi-species gas mixture described by a kinetic model. More precisely, we are interested in models with BGK interaction operators. Several extensions to the standard BGK model are studied.
Firstly, we allow the collision frequency to vary not only in time and space but also with the microscopic velocity. In the standard BGK model, the dependence on the microscopic velocity is neglected for reasons of simplicity. We allow for a more physical description by reintroducing this dependence. But even though the structure of the equations remains the same, the so-called target functions in the relaxation term become more sophisticated being defined by a variational procedure.
Secondly, we include quantum effects (for constant collision frequencies). This approach influences again the resulting target functions in the relaxation term depending on the respective type of quantum particles.
In this thesis, we present a numerical method for simulating such models. We use implicit-explicit time discretizations in order to take care of the stiff relaxation part due to possibly large collision frequencies. The key new ingredient is an implicit solver which minimizes a certain potential function. This procedure mimics the theoretical derivation in the models. We prove that theoretical properties of the model are preserved at the discrete level such as conservation of mass, total momentum and total energy, positivity of distribution functions and a proper entropy behavior. We provide an array of numerical tests illustrating the numerical scheme as well as its usefulness and effectiveness.
Ein Teil der interstellaren Materie (ISM) liegt in Form von winzigen Festkörpern vor, die mit dem interstellaren Gas vermischt sind. Diese Teilchen werden als interstellarer Staub bezeichnet. Obwohl der Staubanteil an der Gesamtmasse der ISM nur etwa 1% beträgt, kann sein Einfluß auf das interstellare Strahlungsfeld und die Dynamik des Gases nicht vernachlässigt werden. So ist er die Hauptursache für Extinktion, Streuung und Polarisation von Licht. Außerdem stellt der Staub ein wichtiges Kühlmittel für das interstellare Medium dar und beeinflußt die chemischen Prozesse innerhalb der ISM. Staubpartikel unterliegen Wachstums- und Zerstörungsprozessen. So können sie Moleküle aus der Umgebung an ihrer Oberfläche anlagern (Akkretion) oder sich mit anderen Partikeln zu größeren Staubteilchen verbinden (Koagulation). Durch die Wechselwirkung mit Ionen kann Oberflächenmaterial abgetragen werden (Sputtering) und das Kollidieren von Staubpartikeln führt zu deren Zerschlagung in kleinere Teilchen oder (Shattering) deren Vaporisation. Außerdem sind Staubpartikel an das Gas gekoppelt und werden von diesem mitgerissen. Der Schwerpunkt der Vorliegenden Arbeit war die Untersuchung der dynamischen Prozesse, denen Staubpartikel bei der Durchquerung von interstellaren Stoßfronten unterworfen sind. In diesem Zusammenhang spielen vorallem die destruktiven Prozesse und die Kopplung an das Gas eine wichtige Rolle. Es wurden Gleichungen eingeführt, die die Änderung einer Staubverteilung durch diese Vorgänge beschreiben. Im Gegensatz zu bisherigen Modellen werden die Staubteilchen darin nicht allein durch ihre Masse, sondern auch durch ihre Geschwindigkeit charakterisiert. Auf diese Weise kann die Impulserhaltung bei einer Partikelkollision gewährleistet werden und es ist beispielsweise möglich auch Stöße gleich schwerer Partikel zu beschreiben. Die Gleichungen der Staub- und Hydrodynamik wurden für den Fall von stationären, eindimensionalen Stoßwellen numerisch gelöst, wobei die Wechselwirkungen zwischen Gas und Staub berücksichtigt wurden. Mit Hilfe des Modells wurden die Wirkung verschieden starker Stoßwellen auf eine Staubverteilung untersucht. Dabei wurden verschiedene Staubmaterialien zugrunde gelegt.
This thesis presents results covering several topics in correlated many fermion systems. A Monte Carlo technique (CT-INT) that has been implemented, used and extended by the author is discussed in great detail in chapter 3. The following chapter discusses how CT-INT can be used to calculate the two particle Green’s function and explains how exact frequency summations can be obtained. A benchmark against exact diagonalization is presented. The link to the dynamical cluster approximation is made in the end of chapter 4, where these techniques are of immense importance. In chapter 5 an extensive CT-INT study of a strongly correlated Josephson junction is shown. In particular, the signature of the first order quantum phase transition between a Kondo and a local moment regime in the Josephson current is discussed. The connection to an experimental system is made with great care by developing a parameter extraction strategy. As a final result, we show that it is possible to reproduce experimental data from a numerically exact CT-INT model-calculation. The last topic is a study of graphene edge magnetism. We introduce a general effective model for the edge states, incorporating a complicated interaction Hamiltonian and perform an exact diagonalization study for different parameter regimes. This yields a strong argument for the importance of forbidden umklapp processes and of the strongly momentum dependent interaction vertex for the formation of edge magnetism. Additional fragments concerning the use of a Legendre polynomial basis for the representation of the two particle Green’s function, the analytic continuation of the self energy for the Anderson Kane Mele Model, as well as the generation of test data with a given covariance matrix are documented in the appendix. A final appendix provides some very important matrix identities that are used for the discussion of technical details of CT-INT.
This thesis contains two major parts: The first part introduces the reader into three independent concepts of treating strongly correlated many body physics. These are, on the analytical side the SO(5)-theory (Chap.3), which poses the general frame. On the numerical side these are the Stochastic Series Expansion (SSE) (Chap.1) and the Contractor Renormalization Group (CORE) approach (Chap. 2}). The central idea of this thesis was to combine these above concepts, in order to achieve a better understanding of the high-T_c superconductors (HTSC). The results obtained by this combination can be found in the second major part of this thesis (chapters 4 and 5). The main idea of this thesis, i.e., to combine the SO(5)-theory with the capabilities of bosonic Quantum-Monte Carlo simulations and those of the CORE approach, has been proven to be a very successful Ansatz. Two different approaches, one based on symmetry and one on renormalization-group arguments, motivate an effective bosonic Hamiltonian. In a subsequent step the effective Hamiltonian has been simulated efficiently using the SSE. The results reproduce salient experiments on high-T_c superconductors. In addition, it has been shown that the model can be extended to capture also charge ordering. These results also form a profound basis for further studies, for example one could address the open question of SO(5)-symmetry restoration at a multicritical point in the extended pSO(5) model, where longer ranged interactions are included.
This thesis, first, is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints, subsequently, as well as constrained structured optimization problems featuring a composite objective function and set-membership constraints. It is then concerned to convergence and rate-of-convergence analysis of proximal gradient methods for the composite optimization problems in the presence of the Kurdyka--{\L}ojasiewicz property without global Lipschitz assumption.
An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.
This thesis is devoted to numerical verification of optimality conditions for non-convex optimal control problems. In the first part, we are concerned with a-posteriori verification of sufficient optimality conditions. It is a common knowledge that verification of such conditions for general non-convex PDE-constrained optimization problems is very challenging. We propose a method to verify second-order sufficient conditions for a general class of optimal control problem. If the proposed verification method confirms the fulfillment of the sufficient condition then a-posteriori error estimates can be computed. A special ingredient of our method is an error analysis for the Hessian of the underlying optimization problem. We derive conditions under which positive definiteness of the Hessian of the discrete problem implies positive definiteness of the Hessian of the continuous problem. The results are complemented with numerical experiments. In the second part, we investigate adaptive methods for optimal control problems with finitely many control parameters. We analyze a-posteriori error estimates based on verification of second-order sufficient optimality conditions using the method developed in the first part. Reliability and efficiency of the error estimator are shown. We illustrate through numerical experiments, the use of the estimator in guiding adaptive mesh refinement.