M.Sc. (Applied Mathematics) / Explicit schemes for integrating ODEs and time–dependent partial differential equations (in the method of lines–MoL–approach) are very well–known to be stable as long as the maximum sizes of their timesteps remain below a certain minimum value of the spatial grid spacing. This is the Courant– Friedrich’s–Lewy (CFL) condition. These schemes are the ones traditionally being used for performing simulations in Numerical Relativity (NR). However, due to the above restriction on the timestep, these schemes tend to be so much inadequate for simulating some of the highly probable and astrophysically interesting phenomenae. So, it is of interest this currernt moment to seek or find integrating schemes that may help numerical relativists to somehow circumvent the CFL restriction inherent in the use of explicit schemes. In this quest, a more natural starting point appears to be implicit schemes. These schemes possess a highly desireable stability property – they are unconditionally stable. There also exists a combination of implicit and explicit (IMEX) schemes. Some researchers have already started exploring (since 2009, 2011) these for NR purposes. We report on the implementation of two implicit schemes (implicit Euler, and implicit midpoint rule) for Einstein’s evolution equations. For low computational costs, we concentrated on spherical symmetry. The integration schemes were successfully implemented and showed satisfactory second order convergence patterns on the systems considered. In particular, the Implicit Midpoint Rule proved to be a little superior to the implicit Euler scheme.

Abstract available on page 4 of the document / Thesis (PhD (Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

We introduce a new framework of generalized operators to handle vector valued distributions, intertwined evolution operators of B-evolution equations and Fokker Planck type evolution equations. Generalized operators capture these operators. The framework is a marriage between vector valued distribution theory and abstract harmonic analysis: a new convolution algebra is the offspring. The new algebra shows that convolution is more fundamental than operator composition. The framework is complete with a Hille-Yosida theorem for implicit evolution equations for generalized operators. Feller semigroups and processes fit perfectly into the framework of generalized operators. Feller semigroups are intertwined by the Chapman Kolmogorov equation. Our framework handles more complex intertwinements which naturally arise from a dynamic boundary approach to an absorbing barrier of a fly trap model: we construct an entwined pseudo Poisson process which is a pair of stochastic processes entwined by the extended Chapman Kolmogorov equation. Similarly, we introduce the idea of an entwined Brownian motion. We show that the diffusion equation of an entwined Brownian motion involves an implicit evolution equation on a suitable scalar test space. We end off by constructing a new convolution of operator valued measures which generalizes the convolution of Feller convolution semigroups. / Thesis (PhD)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / Unrestricted

The present work in the area of soliton theory studies two problems which arise when seeking analytic solutions to certain nonlinear partial differential equations. In the first one, Lax pairs associated with prolonged eigenfunctions and prolonged squared eigenfunctions (prolonged squares) are derived for a Schrödinger equation with a potential depending polynomially on the spectral parameter (of degree N) and its respective hierarchy of nonlinear evolution equations (here named generalized polynomial Korteweg-de Vries equations or GPKdV). It is shown that the prolonged squares satisfy the linearized GPKdV equations. On that basis, the Cauchy problem for the linearized GPKdV has been solved by finding a complete set of such prolonged squares and applying an expansion formula derived in another work by the author. The results are a generalization of the ones by Sachs (SIAM J. Math. Anal. 14, 1983, 674-683). Moreover, a condition on the so-called recursion operator A is derived which generates the whole hierarchy of Lax pairs associated with the prolonged squares. As for the second part of the work, it developed a scheme for deriving gauge transformations between different linear spectral problems. Then the scheme is applied to obtain all known Darboux transformations for a quadratic pencil (the spectral problem considered in the first part at N = 2), Schrödinger equation (N = 1), Ablowitz-Kaup-Newell-Segur (AKNS) system and also derive the Jaulent-Miodek transformation. Moreover, the scheme yields a large family of new transformations of the above types. It also gives some insight on the structure of the transformations and emphasizes the symmetry with respect to inversion that they possess. / Ph. D.

Orientador: Wilson Castro Ferreira Junior / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T05:05:37Z (GMT). No. of bitstreams: 1 Assis_RaulAbreude_D.pdf: 6617583 bytes, checksum: 80b80410952f2ac47edc7cf71466fb07 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho apresentamos modelos matemáticos de processos evolutivos, utilizando como abordagem principal a descrição da frequência de fenótipos em populações. São propostos diversos modelos baseados em equações diferenciais parciais, ordinárias e equações de recorrência. São apresentados resultados de suas análises, bem como comparações com modelos genéticos aditivos. Como uma ilustração do tipo de abordagem proposto, criamos um modelo de um sistema do tipo parasita-hospedeiro (Maculinea-Myrmica) utilizando-o com o objetivo de analisar comportamento de especificidade de hospedeiro. Finalmente, apresentamos generalizações dos modelos em espaços de fenótipos e indicamos direções para aprofundamento de pesquisa / Abstract: In this thesis we present models of evolutionary dynamics that describe the changes in frequencies of phenotypes in populations. The models proposed are based on ordinary and partial differential equations and difference equations. Results from the analysis of the models and comparisons with the behavior of additive genetic models are presented. We develop a model for a host-parasite system (Maculinea-Myrmica) using the phenotypic space aprroach with the objective of analysing the hostspecificity behavior of the species. Finally, we present generalizations of the models of phenotypic spaces and indicate directions for further research in the area / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Cette thèse présente des travaux originaux dans le domaine des méthodes de moyennisation d'ordre élevé. On s'intéresse notamment à des procédures de moyennisation dite stroboscopique ou quasi-stroboscopique dans des espaces de Banach ou de Hilbert. Ces procédures sont ensuite appliquées à des exemples concrets: des équations d'évolutions hautement oscillantes. Plus précisément, on montre dans un premier temps un résultat de moyennisation stroboscopique dans un espace de Banach où l'on obtient des estimations d'erreurs exponentielles. Ce théorème est ensuite appliqué sur deux équations des ondes semi-linéaire hautement oscillantes. On montre également que la Stroboscopic Averaging Method s'applique à une équation des ondes semi-linéaire avec conditions de Dirichlet. On trouve enfin numériquement, une dynamique intéressante de l'équation des ondes semi-linéaire mise en lumière par la procédure de moyennisation. Dans un second temps, on présente un théorème de moyennisation quasi-stroboscopique dans un espace de Hilbert quelconque avec des estimations d'erreurs exponentielles. Ce théorème est alors appliqué de façon indirecte à une équation de Schrödinger semi-linéaire oscillante. Cette équation est d'abord projeté dans un espace de dimension finie pour qu'on puisse lui appliquer le théorème de moyennisation quasi-stroboscopique. On écrit alors un résultat de moyennisation quasi-stroboscopique pour l'équation de Schrödinger semi-linéaire avec des estimations d'erreur polynomiales. / This thesis presents some original work in the field of high order averaging procedure. In particular, we are interested in stroboscopic and quasi-stroboscopic averaging procedure in abstract Banach or Hilbert spaces. This procedures is applied to concrete examples: some highly oscillatory evolution equations. More precisely, we first show a theorem of stroboscopic averaging in a Banach space where we obtain exponential error estimates. This theorem is then applied on two semi-linear and highly oscillatory wave equations. We also put in evidence that the {\it Stroboscopic Averaging Method} works fine with a semi-linear wave equation with Dirichlet conditions. Finally, the averaging procedure puts in evidence, numerically, an interesting dynamics regarding the semi-linear wave equation with Dirichlet conditions. In a second part, we present a quasi-stroboscopic averaging theorem in a Hilbert space with exponential error estimates. This theorem is applied on a semi-linear Schrödinger equation. This equation has first, to be project in a finite dimensional space in order to fit in the hypotheses of the theorem. We then write a quasi-stroboscopic averaging theorem for a semi-linear Schrödinger equation with polynomial error estimates.

Dans cette thèse, nous étudions la stabilisation de certaines équations d’évolution par des lois de rétroaction. Dans le premier chapitre nous étudions l’équation des ondes dans R avec conditions aux limites dynamiques appliquées sur une partie du bord et une condition de Dirichlet sur la partie restante. Nous fournissons des conditions suffisantes qui garantissent une stabilité polynomiale en utilisant une méthode qui combine une inégalité d’observabilité pour le problème non amorti associé avec des résultats de régularité du problème non amorti. L’optimalité de la décroissance est montrée dans certains cas à l’aide des résultats spectraux précis de l’opérateur associé. Dans le deuxième chapitre nous considérons le système sur un domaine de Rd, d ≥ 2. On trouve des conditions suffisantes qui permettent la stabilité forte. Ensuite, nous discutons de la stabilité non uniforme ainsi que de la stabilité polynomiale. L’approche en domaine fréquentiel nous permet d’établir une décroissance polynomiale sur des domaines pour lesquels l’équation des ondes avec l’amortissement standard est exponentiellement ou polynomialement stable. Dans le troisième chapitre nous considérons un cadre général d’équations d’évolution avec une dissipation dynamique. Sous une hypothèse de régularité, nous montrons que les propriétés d’observabilité pour le problème non amorti impliquent des estimations de décroissance pour le problème amorti. / In this thesis, we study the stabilization of some evolution equations by feedback laws. In the first chapter we study the wave equation in R with dynamical boundary control applied on a part of the boundary and a Dirichlet boundary condition on the remaining part. We furnish sufficient conditions that guarantee a polynomial stability proved using a method that combines an observability inequality for the associated undamped problem with regularity results of the solution of the undamped problem. In addition, the optimality of the decay is shown in some cases with the help of precise spectral results of the operator associated with the damped problem. Then in the second chapter we consider the system on a domain of Rd, d ≥ 2. In this case, the domain of the associated operator is not compactly embedded into the energy space. Nevertheless, we find sufficient conditions that give the strong stability. Then, we discuss the non uniform stability as well as the polynomial stability by two methods. The frequency domain approach allows us to establish a polynomial decay on some domains for which the wave equation with the standard damping is exponentially or polynomially stable. Finally, in the third chapter we consider a general framework of second order evolution equations with dynamical feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We finally illustrate our general results by a variety of examples.

Diese Dissertation beschäftigt sich mit EDP-Konvergenz. Dabei handelt es sich um einen Konvergenzbegriff auf dem Gebiet der verallgemeinerten Gradientensysteme und metrischen Gradientensysteme, der geeignet ist für Gradientenflüsse, die von einem kleinen Parameter abhängen. EDP-Konvergenz liefert einen Algorithmus, der es erlaubt in der Energie und dem Dissipationspotenzial zum Grenzwert überzugehen. Es ist die fundamentale Frage evolutionärer Γ-Konvergenz, wie das Limes-Dissipationspotenzial berechnet werden kann. Das Ziel dieser Arbeit ist es aufzuzeigen, dass EDP-Konvergenz das mikro- und das makroskopische Dissipationspotenzial in einer sinnvollen und eindeutigen Art und Weise in Beziehung setzt. Anhand von drei Beispielen wird der Konvergenzbegriff untersucht: die Diffusionsgleichung auf einem dünnen, dreischichtigen Gebiet, die Poröse-Medien-Gleichung mit einer dünnen Membran und ein Modell mit oszillierender Energie. Es wird die Definition von relaxierter EDP-Konvergenz und EDP-Konvergenz mit Kippung motiviert. EDP-Konvergenz basiert auf dem Prinzip, dass es ein Gleichgewicht zwischen Energie und Dissipation gibt – das Energie-Dissipations-Prinzip (EDP). Mittels Γ-Konvergenz wird sowohl in der Energie, als auch dem totalen Dissipationsfunktional zum Grenzwert übergegangen. Durch die zusätzliche Entkopplung von Zustand und Triebkraft wird die Dissipationslandschaft erkundet und die kinetische Beziehung des Limessystems ermittelt. Das Modell mit oszillierender Energie zeigt die Bedeutung der kinetischen Beziehung – und damit der Kippung – für die Herleitung des Limes-Dissipationspotenzials auf. Die Modelle mit Wasserstein-Dissipation zeigen, dass das Limes-Dissipationspotenzial nicht der naive Grenzwert ist. Insbesondere können klassische Gradientensysteme mit quadratischer Dissipation zu verallgemeinerten Gradientensysteme konvergieren. / In the realm of generalized gradient systems and metric gradient systems we study a notion of convergence suited for gradient flows which depend on a small parameter. This notion is called EDP-convergence. In order to understand the convergence of gradient systems we need an algorithm to derive the limiting energy as well as the limiting dissipation potential. The fundamental question of evolutionary Γ-convergence is how to compute the limit dissipation potential. The aim of this thesis is to show that EDP-convergence connects the microscopic dissipation potential with the macroscopic, i.e. limiting, dissipation potential in a meaningful and unique way. As a proof of concept 3 different examples are presented: (i) the diffusion equation on a thin sandwich-like domain, (ii) the porous medium equation with a thin interface and (iii) a wiggly energy model. We show how the gradient flow concept that is used in this thesis can be used to obtain also gradient flows with respect to the Wasserstein metric. We motivate the definition of relaxed EDP-convergence and EDP- convergence with tilting. EDP-convergence is based upon the principle that there is an energy-dissipation-balance involving the total dissipation functional and the energy difference – the energy-dissipation-principle (EDP). The limit passage, in both the energy and the total dissipation functional, is performed in terms of Γ-convergence. By perturbing the flow as well as the driving force, the dissipation-landscape is explored and a kinetic relation for the limit system can be established. The wiggly energy model demonstrates the importance of the kinetic relation for the construction of the limiting dissipation potential and thus the introduction of tilts. The models with a Wasserstein dissipation show that the limiting dissipation potential is not the naive limit. In particular, classical gradient systems with a quadratic dissipation potential converge to a generalized gradient systems.

In a Hilbert space setting homogenization of evolutionary equations is discussed. In order to do so, a suitable topology on material laws is introduced and several properties of that topology are shown. With those properties homogenization theorems of a large class of linear evolutionary problems of classical mathematical physics can be obtained. The results are exemplified by the equations of piezo-electro-magnetism.

This dissertation considers several problems related to Ricci flow, including the existence and behavior of solutions. The first goal is to obtain explicit, coordinate-based descriptions of Ricci flow solutions--especially those corresponding to Ricci solitons--on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. Next, we consider solsolitons involving the nilsolitons in the Heisenberg case above. This uses work of Lauret, which characterizes solsolitons as certain extensions of nilsolitons, and work of Will, which demonstrates that the space of solsolitons extensions of a given nilsoliton is parametrized by the quotient of a Grassmannian by a finite group. We determine these spaces of solsoliton extensions of Heisenberg nilsolitons, and we also explicitly describe many-parameter families of these solsolitons in dimensions greater than three. Finally, we explore Ricci flow coupled with harmonic map flow, both as it arises naturally in certain bundle constructions related to Ricci flow and as a geometric flow in its own right. In the first case, we generalize a theorem of Knopf that demonstrates convergence and stability of certain locally R[superscript N]-invariant Ricci flow solutions. In the second case, we prove a version of Hamilton's compactness theorem for the coupled flow, and then generalize it to the category of etale Riemannian groupoids. We also provide a detailed example of solutions to the flow on the three-dimensional Heisenberg group. / text