@phdthesis{Warnecke2022,
  author    = {Warnecke, Sandra},
  title     = {Numerical schemes for multi-species BGK equations based on a variational procedure applied to multi-species BGK equations with velocity-dependent collision frequency and to quantum multi-species BGK equations},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-192-1},
  doi       = {10.25972/WUP-978-3-95826-193-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-282378},
  school      = {W{\"u}rzburg University Press},
  pages     = {xiii, 203},
  year      = {2022},
  abstract  = {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.},
  subject      = {Kinetische Gastheorie},
  language  = {en}
}
@phdthesis{GallegoValencia2017,
  author    = {Gallego Valencia, Juan Pablo},
  title     = {On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-148874},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {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.},
  subject      = {Eulersche Differentialgleichung},
  language  = {en}
}
@phdthesis{Akindeinde2012,
  author    = {Akindeinde, Saheed Ojo},
  title     = {Numerical Verification of Optimality Conditions in Optimal Control Problems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-76065},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {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.},
  subject      = {Optimale Kontrolle},
  language  = {en}
}
@phdthesis{Luitz2012,
  author    = {Luitz, David J.},
  title     = {Numerical methods and applications in many fermion systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-75927},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {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.},
  subject      = {Fermionensystem},
  language  = {en}
}
@phdthesis{Joestingmeier2005,
  author    = {J{\"o}stingmeier, Martin},
  title     = {On the competition of superconductivity, antiferromagnetism and charge order in the high-Tc compounds},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-13036},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2005},
  abstract  = {Diese Arbeit l{\"a}ßt sich in zwei grobe Abschnitte gliedern. Der erste Teil umfaßt die Kapitel 1-3, in denen drei verschiedene Konzepte beschrieben werden, die zum Verst{\"a}ndis stark korrelierter Vielteilchen-Systeme dienen. Dies sind zun{\"a}chst einmal die SO(5)-Theorie in Kapitel 3, die den allgemeinen Rahmen vorgibt und auf der numerischen Seite die Stochastische Reihen Entwicklung (SSE) in Kapitel 1 und der Contractor Renormierungsgruppen Ansatz (CORE), s.Kapitel 2). Die zentrale Idee dieser Dissertationsschrift besteht darin, diese verschiedenen Konzepte zu kombinieren, um ein besseres Verst{\"a}ndnis der Hochtemperatursupraleiter zu erhalten. Im zweiten Teil dieser Arbeit (Kap. 4 und Kap. 5) werden die so gewonnenen Ergebnisse dargestellt. Die zentrale Idee dieser Arbeit, d.h. die Kombination der SO(5)-Theorie mit den F{\"a}higkeiten bosonischer Quanten-Monte-Carlo Verfahren und den {\"u}berlegungen der Renormierungsgruppe, hat sich sich am Beispiel der Physik der Hochtemperatur-Supraleiter als sehr tragf{\"a}hig erwiesen. Die numerischen Simulationen reproduzieren bei den behandelten Modelle eine Reihe wichtiger experimenteller Daten. Die Grundlage f{\"u}r eine k{\"u}nftige weitere schrittweise Erweiterung des Modells wurde so geschaffen. Eine offene Frage ist z.B. die Restaurierung der SO(5)-Symmetrie an einem multi-kritischen Punkt, wenn die l{\"a}ngerreichweitigen Wechselwirkungen mit in das Modell einbezogen sind.},
  subject      = {Hochtemperatursupraleitung},
  language  = {en}
}
@phdthesis{Voelker2003,
  author    = {V{\"o}lker, Roland},
  title     = {Staubzerst{\"o}rung durch interstellare Stoßfronten},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-7707},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2003},
  abstract  = {Ein Teil der interstellaren Materie (ISM) liegt in Form von winzigen Festk{\"o}rpern 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\% betr{\"a}gt, kann sein Einfluß auf das interstellare Strahlungsfeld und die Dynamik des Gases nicht vernachl{\"a}ssigt werden. So ist er die Hauptursache f{\"u}r Extinktion, Streuung und Polarisation von Licht. Außerdem stellt der Staub ein wichtiges K{\"u}hlmittel f{\"u}r das interstellare Medium dar und beeinflußt die chemischen Prozesse innerhalb der ISM. Staubpartikel unterliegen Wachstums- und Zerst{\"o}rungsprozessen. So k{\"o}nnen sie Molek{\"u}le aus der Umgebung an ihrer Oberfl{\"a}che anlagern (Akkretion) oder sich mit anderen Partikeln zu gr{\"o}ßeren Staubteilchen verbinden (Koagulation). Durch die Wechselwirkung mit Ionen kann Oberfl{\"a}chenmaterial abgetragen werden (Sputtering) und das Kollidieren von Staubpartikeln f{\"u}hrt 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 eingef{\"u}hrt, die die {\"A}nderung einer Staubverteilung durch diese Vorg{\"a}nge 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 gew{\"a}hrleistet werden und es ist beispielsweise m{\"o}glich auch St{\"o}ße gleich schwerer Partikel zu beschreiben. Die Gleichungen der Staub- und Hydrodynamik wurden f{\"u}r den Fall von station{\"a}ren, eindimensionalen Stoßwellen numerisch gel{\"o}st, wobei die Wechselwirkungen zwischen Gas und Staub ber{\"u}cksichtigt wurden. Mit Hilfe des Modells wurden die Wirkung verschieden starker Stoßwellen auf eine Staubverteilung untersucht. Dabei wurden verschiedene Staubmaterialien zugrunde gelegt.},
  subject      = {Interstellarer Staub},
  language  = {de}
}