Refine
Has Fulltext
- yes (14)
Is part of the Bibliography
- yes (14)
Year of publication
- 2018 (14) (remove)
Document Type
- Doctoral Thesis (10)
- Journal article (2)
- Other (2)
Language
- English (14)
Keywords
- Code examples (2)
- SQH method (2)
- *-algebra (1)
- Acoustic equations (1)
- Asymptotic Preserving (1)
- Atmosphäre (1)
- Banach-Raum (1)
- Beatty sequence (1)
- Bregman distance (1)
- Burgers-Gleichung (1)
Institute
- Institut für Mathematik (14) (remove)
This work is concerned with the numerical approximation of solutions to models that are used to describe atmospheric or oceanographic flows. In particular, this work concen- trates on the approximation of the Shallow Water equations with bottom topography and the compressible Euler equations with a gravitational potential. Numerous methods have been developed to approximate solutions of these models. Of specific interest here are the approximations of near equilibrium solutions and, in the case of the Euler equations, the low Mach number flow regime. It is inherent in most of the numerical methods that the quality of the approximation increases with the number of degrees of freedom that are used. Therefore, these schemes are often run in parallel on big computers to achieve the best pos- sible approximation. However, even on those big machines, the desired accuracy can not be achieved by the given maximal number of degrees of freedom that these machines allow. The main focus in this work therefore lies in the development of numerical schemes that give better resolution of the resulting dynamics on the same number of degrees of freedom, compared to classical schemes.
This work is the result of a cooperation of Prof. Klingenberg of the Institute of Mathe- matics in Wu¨rzburg and Prof. R¨opke of the Astrophysical Institute in Wu¨rzburg. The aim of this collaboration is the development of methods to compute stellar atmospheres. Two main challenges are tackled in this work. First, the accurate treatment of source terms in the numerical scheme. This leads to the so called well-balanced schemes. They allow for an accurate approximation of near equilibrium dynamics. The second challenge is the approx- imation of flows in the low Mach number regime. It is known that the compressible Euler equations tend towards the incompressible Euler equations when the Mach number tends to zero. Classical schemes often show excessive diffusion in that flow regime. The here devel- oped scheme falls into the category of an asymptotic preserving scheme, i.e. the numerical scheme reflects the behavior that is computed on the continuous equations. Moreover, it is shown that the diffusion of the numerical scheme is independent of the Mach number.
In chapter 3, an HLL-type approximate Riemann solver is adapted for simulations of the Shallow Water equations with bottom topography to develop a well-balanced scheme. In the literature, most schemes only tackle the equilibria when the fluid is at rest, the so called Lake at rest solutions. Here a scheme is developed to accurately capture all the equilibria of the Shallow Water equations. Moreover, in contrast to other works, a second order extension is proposed, that does not rely on an iterative scheme inside the reconstruction procedure, leading to a more efficient scheme.
In chapter 4, a Suliciu relaxation scheme is adapted for the resolution of hydrostatic equilibria of the Euler equations with a gravitational potential. The hydrostatic relations are underdetermined and therefore the solutions to that equations are not unique. However, the scheme is shown to be well-balanced for a wide class of hydrostatic equilibria. For specific classes, some quadrature rules are computed to ensure the exact well-balanced property. Moreover, the scheme is shown to be robust, i.e. it preserves the positivity of mass and energy, and stable with respect to the entropy. Numerical results are presented in order to investigate the impact of the different quadrature rules on the well-balanced property.
In chapter 5, a Suliciu relaxation scheme is adapted for the simulations of low Mach number flows. The scheme is shown to be asymptotic preserving and not suffering from excessive diffusion in the low Mach number regime. Moreover, it is shown to be robust under certain parameter combinations and to be stable from an Chapman-Enskog analysis.
Numerical results are presented in order to show the advantages of the new approach.
In chapter 6, the schemes developed in the chapters 4 and 5 are combined in order to investigate the performance of the numerical scheme in the low Mach number regime in a gravitational stratified atmosphere. The scheme is shown the be well-balanced, robust and stable with respect to a Chapman-Enskog analysis. Numerical tests are presented to show the advantage of the newly proposed method over the classical scheme.
In chapter 7, some remarks on an alternative way to tackle multidimensional simulations are presented. However no numerical simulations are performed and it is shown why further research on the suggested approach is necessary.
Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields.
Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces
(2018)
This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces.
A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting.
The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.
This thesis discusses and proposes a solution for one problem arising from deformation quantization:
Having constructed the quantization of a classical system, one would like to understand its mathematical properties (of both the classical and quantum system). Especially if both systems are described by ∗-algebras over the field of complex numbers, this means to understand the properties of certain ∗-algebras:
What are their representations? What are the properties of these representations? How
can the states be described in these representations? How can the spectrum of the observables be
described?
In order to allow for a sufficiently general treatment of these questions, the concept of abstract O ∗-algebras is introduced. Roughly speaking, these are ∗ -algebras together with a cone of positive linear functionals on them (e.g. the continuous ones if one starts with a ∗-algebra that is endowed with a well-behaved topology). This language is then applied to two examples from deformation quantization, which will be studied in great detail.
The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of Gamma-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density. Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions. A crucial ingredient to our proofs is a novel decomposition of the energy that allows for re fined estimates.
In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.
Ill-posed optimization problems appear in a wide range of mathematical applications, and their numerical solution requires the use of appropriate regularization techniques. In order to understand these techniques, a thorough analysis is inevitable.
The main subject of this book are quadratic optimal control problems subject to elliptic linear or semi-linear partial differential equations. Depending on the structure of the differential equation, different regularization techniques are employed, and their analysis leads to novel results such as rate of convergence estimates.
The present thesis considers the modelling of gas mixtures via a kinetic description. Fundamentals about the Boltzmann equation for gas mixtures and the BGK approximation are presented. Especially, issues in extending these models to gas mixtures are discussed. A non-reactive two component gas mixture is considered. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the two species. The model presented here contains several models from physicists and engineers as special cases. Consistency of this model is proven: conservation properties, positivity of all temperatures and the H-theorem. The form in global equilibrium as Maxwell distributions is specified. Moreover, the usual macroscopic conservation laws can be derived.
In the literature, there is another type of BGK model for gas mixtures developed by Andries, Aoki and Perthame, which contains only one interaction term. In this thesis, the advantages of these two types of models are discussed and the usefulness of the model presented here is shown by using this model to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described in the literature by Dellacherie. In addition, for each of the two models existence and uniqueness of mild solutions is shown. Moreover, positivity of classical solutions is proven.
Then, the model presented here is applied to three physical applications: a plasma consisting of ions and electrons, a gas mixture which deviates from equilibrium and a gas mixture consisting of polyatomic molecules.
First, the model is extended to a model for charged particles. Then, the equations of magnetohydrodynamics are derived from this model. Next, we want to apply this extended model to a mixture of ions and electrons in a special physical constellation which can be found for example in a Tokamak. The mixture is partly in equilibrium in some regions, in some regions it deviates from equilibrium. The model presented in this thesis is taken for this purpose, since it has the advantage to separate the intra and interspecies interactions. Then, a new model based on a micro-macro decomposition is proposed in order to capture the physical regime of being partly in equilibrium, partly not. Theoretical results are presented, convergence rates to equilibrium in the space-homogeneous case and the Landau damping for mixtures, in order to compare it with numerical results.
Second, the model presented here is applied to a gas mixture which deviates from equilibrium such that it is described by Navier-Stokes equations on the macroscopic level. In this macroscopic description it is expected that four physical coefficients will show up, characterizing the physical behaviour of the gases, namely the diffusion coefficient, the viscosity coefficient, the heat conductivity and the thermal diffusion parameter. A Chapman-Enskog expansion of the model presented here is performed in order to capture three of these four physical coefficients. In addition, several possible extensions to an ellipsoidal statistical model for gas mixtures are proposed in order to capture the fourth coefficient. Three extensions are proposed: An extension which is as simple as possible, an intuitive extension copying the one species case and an extension which takes into account the physical motivation of the physicist Holway who invented the ellipsoidal statistical model for one species. Consistency of the extended models like conservation properties, positivity of all temperatures and the H-theorem are proven. The shape of global Maxwell distributions in equilibrium are specified.
Third, the model presented here is applied to polyatomic molecules. A multi component gas mixture with translational and internal energy degrees of freedom is considered. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic ellipsoidal statistical equations. Consistency of this model is shown: conservation properties, positivity of the temperature, H-theorem and the form of Maxwell distributions in equilibrium. For numerical purposes the Chu reduction is applied to the developed model for polyatomic gases to reduce the complexity of the model and an application for a gas consisting of a mono-atomic and a diatomic gas is given.
Last, the limit from the model presented here to the dissipative Euler equations for gas mixtures is proven.
This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant.
\begin{align*}
\partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t))
\end{align*}
The model was introduced for the case of a single particle by Lagoutière, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves.
The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered.
The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling.
Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs.
Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated.
A mathematical optimal-control tumor therapy framework consisting of radio- and anti-angiogenesis control strategies that are included in a tumor growth model is investigated. The governing system, resulting from the combination of two well established models, represents the differential constraint of a non-smooth optimal control problem that aims at reducing the volume of the tumor while keeping the radio- and anti-angiogenesis chemical dosage to a minimum. Existence of optimal solutions is proved and necessary conditions are formulated in terms of the Pontryagin maximum principle. Based on this principle, a so-called sequential quadratic Hamiltonian (SQH) method is discussed and benchmarked with an “interior point optimizer―a mathematical programming language” (IPOPT-AMPL) algorithm. Results of numerical experiments are presented that successfully validate the SQH solution scheme. Further, it is shown how to choose the optimisation weights in order to obtain treatment functions that successfully reduce the tumor volume to zero.