The production of axisymmetric initial data for distorted black holes at a moment of time symmetry is considered within the (3+1) context of general relativity. The initial data is made to contain a distorted marginally trapped surface ensuring that, modulo cosmic censorship, the spacetime will contain a black hole. The resulting equations on the complicated domain are solved using the piecewise linear finite element method which adapts to the curved surface of the marginally trapped surface. The initial data is then analyzed to calculate the mass of the space time as well as an upper bound on the fraction of the total energy available for radiation. The families of initial data considered contain no more than few percent of the total energy available for radiation even in cases of extreme distortion. It is shown that the mass of certain initial data slices depend to first order on the area of the marginally trapped surface and the gaussian curvature of prominent features.

The production of axisymmetric initial data for distorted black holes at a moment of time symmetry is considered within the (3+1) context of general relativity. The initial data is made to contain a distorted marginally trapped surface ensuring that, modulo cosmic censorship, the spacetime will contain a black hole. The resulting equations on the complicated domain are solved using the piecewise linear finite element method which adapts to the curved surface of the marginally trapped surface. The initial data is then analyzed to calculate the mass of the space time as well as an upper bound on the fraction of the total energy available for radiation. The families of initial data considered contain no more than few percent of the total energy available for radiation even in cases of extreme distortion. It is shown that the mass of certain initial data slices depend to first order on the area of the marginally trapped surface and the gaussian curvature of prominent features.

The production of axisymmetric initial data for distorted black holes at a moment of time symmetry is considered within the (3+1) context of general relativity. The initial data is made to contain a distorted marginally trapped surface ensuring that, modulo cosmic censorship, the spacetime will contain a black hole. The resulting equations on the complicated domain are solved using the piecewise linear finite element method which adapts to the curved surface of the marginally trapped surface. The initial data is then analyzed to calculate the mass of the space time as well as an upper bound on the fraction of the total energy available for radiation. The families of initial data considered contain no more than few percent of the total energy available for radiation even in cases of extreme distortion. It is shown that the mass of certain initial data slices depend to first order on the area of the marginally trapped surface and the gaussian curvature of prominent features. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In General Relativity, the Einstein field equations allow us to study the evolution of a spacelike 3-manifold, provided that its metric and extrinsic curvature satisfy a system of geometric constraint equations. The so-called Einstein constraint equations, arise as a consequence of the fact that the 3-manifold in question is necessarily a submanifold of the spacetime its evolution defines. This thesis is devoted to a study of the structure of the Einstein constraint system in the special case when the spacelike 3-manifold also satisfies the quasispherical ansatz of Bartnik [B93]. We make no mention of the generality of this gauge; the extent to which the quasispherical ansatz applies remains an open problem. After imposing the quasispherical gauge, we give an argument to show that the resulting Einstein constraint system may be viewed as a coupled system of partial differential equations for the parameters describing the metric and second fundamental form. The hencenamed quasisperical Einstein constraint system, consists of a parabolic equation, a first order elliptic system and (essentially) a system of ordinary differential equations. The question of existence of solutions to this system naturally arises and we provide a partial answer to this question. We give conditions on the initial data and prescribable fields under which we may conclude that the quasispherical Einstein constraint system is uniquley solvable, at least in a region surrounding the unit sphere. The proof of this fact is centred on a linear iterative system of partial differential equations, which also consist of a parabolic equation, a first order elliptic system and a system of ordinary differential equations. We prove that this linear system consistently defines a sequence, and show via a contraction mapping argument, that this sequence must converge to a fixed point of the iteration. The iteration, however, has been specifically designed so that any fixed point of the iteration coincides with a solution of the quasispherical Einstein constraints. The contraction mapping argument mentioned above, relies heavily on a priori estimates for the solutions of linear parabolic equations. We generalise and extend known results 111 concerning parabolic equations to establish special a priori estimates which relate a useful property: the L2-Sobolev regularity of the solution of a parabolic equation is greater than that of the coefficients of the elliptic operator, provided that the initial data is sufficiently regular. This 'smoothing' property of linear parabolic equations along with several estimates from elliptic and ordinary differential equation theory form the crucial ingredients needed in the proof of the existence of a fixed point of the iteration. We begin in chapter one by giving a brief review of the extensive literature concerning the initial value problem in General Relativity. We go on, after mentioning two of the traditional methods for constructing spacetime initial data, to introduce the notion of a quasispherical foliation of a 3-manifold and present the Einstein constraint system written in terms of this gauge. In chapter two we introduce the various inequalities and tracts of analysis we will make use of in subsequent chapters. In particular we define the, perhaps not so familiar, complex differential operator 9 (edth) of Newman and Penrose. In chapter three we develop the appropriate Sobolev-regularity theory for linear parabolic equations required to deal with the quasispherical initial data constraint equations. We include a result due to Polden [P] here, with a corrected proof. This result was essential for deriving the results contained in the later chapters of [P], and it is for this reason we include the result. We don't make use of it explicitly when considering the quasispherical Einstein constraints, but the ideas employed are similar to those we use to tackle the problem of existence for the quasispherical constraints. Chapter four is concerned with the local existence of quasispherical initial data. We firstly consider the question of existence and uniqueness when the mean curvature of the 3-manifold is prescribed, then after introducing the notion of polar curvature, we also present another quasispherical constraint system in which we consider the polar curvature as prescribed. We prove local existence and uniqueness results for both of these alternate formulations of the quasispherical constraints. This thesis was typeset using LATEXwith the package amssymb.

Diese Doktorarbeit behandelt neue Methoden der numerischen Evolution von Systemen mit binären Schwarzen Löchern. Wir analysieren und vergleichen Evolutionen von verschiedenen physikalisch motivierten Anfangsdaten und zeigen Resultate der ersten Evolution von so genannten 'Thin Sandwich' Daten, die von der Gruppe in Meudon entwickelt wurden. <br /> Zum ersten Mal wurden zwei verschiedene Anfangsdaten anhand von dreidimensionalen Evolutionen verglichen: die Puncture-Daten und die Thin-Sandwich Daten. Diese zwei Datentypen wurden im Hinblick auf die physikalischen Eigenschaften während der Evolution verglichen. <br /> Die Evolutionen zeigen, dass die Meudon Daten im Vergleich zu Puncture Daten wesentlich mehr Zeit benötigen bevor sie kollidieren. Dies deutet auf eine bessere Abschätzung der Parameter hin. Die Kollisionszeiten der numerischen Evolutionen sind konsistent mit unabhängigen Schätzungen basierend auf Post-Newtonschen Näherungen die vorhersagen, dass die Schwarzen Löcher ca. 60% eines Orbits rotieren bevor sie kollidieren. / This thesis presents new approaches to evolutions of binary black hole systems in numerical relativity. We analyze and compare evolutions from various physically motivated initial data sets, in particular presenting the first evolutions of Thin Sandwich data generated by the Meudon group. <br /> For the first time two different quasi-circular orbit initial data sequences are compared through fully 3d numerical evolutions: Puncture data and Thin Sandwich data (TSD) based on a helical killing vector ansatz. The two different sets are compared in terms of the physical quantities that can be measured from the numerical data, and in terms of their evolutionary behavior. <br /> The evolutions demonstrate that for the latter, "Meudon" datasets, the black holes do in fact orbit for a longer amount of time before they merge, in comparison with Puncture data from the same separation. This indicates they are potentially better estimates of quasi-circular orbit parameters. The merger times resulting from the numerical simulations are consistent with independent Post-Newtonian estimates that the final plunge phase of a black hole inspiral should take 60% of an orbit.

For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada

We show our work to refine the process of evolutions in general relativistic magnetohydrodynamics. We investigate several areas in order to improve the overall accuracy of our results. We test several versions of conversion methodologies between different sets of variables. We compare both single equation and two equations solvers to do the conversion. We find no significant improvement for multiple equation conversion solvers when compared to single equation solvers. We also investigate the construction of initial data and the conversion of coordinate systems between initial data code and evolution code. In addition to the conversion work, we have improved some methodologies to ensure data integrity when moving data from the initial data code to the evolution code. Additionally we add into the system of MHD equations a new field to help control the no monopole constraint. We perform a characteristic decomposition of the system of equations in order to derive the associated boundary condition for this new field. Finally, we implement a WENO (weighted non-oscillatory) system. This is done so we can evolve and track shocks that are generated during an evolution of our GRMHD equations.

Dans cette thèse, on a d’abord considéré une équation d'onde. On a premièrement montré que l’équation est bien-posée presque sûre par la méthode de décomposition de fréquence de Bourgain sous l’hypothèse de régularité que s > 2(p−3)/(p-1). Ensuite, nous avons réduit de cette exigence de régulation à (p-3)/(p−1) en appelant une estimation probabiliste a priori. Nous considérons également l’approximation des solutions obtenues ci-dessus par des solutions lisses et la stabilité de cette procédure d’approximation. Et nous avons conclu que l’équation est partout mal-posée dans le régime de super-critique. Nous avons considéré ensuite l’équation du faisceau quintique sur le tore 3D. Et nous avons montré que cette équation est presque sûr bien-posée globalement dans certain régimes de super-critique. Enfin, nous avons prouvé que la mesure de l’image de la mesure gaussienne sous l’application de flot de l’équation BBM généralisé satisfait une inégalité de type log-Sobolev avec une petit peu de perte de l’intégrabilité. / In this thesis, we consider a wave equation. We first showed that the equation is almost sure global well-posed via Bourgain’s high-low frequency decomposition under the regularity assumption s > 2(p−3)/(p−1). Then we lowered down this regularity requirement to be (p−3)/(p−1) by invoking a probabilistic a priori estimate. We also consider approximation of the above achieved solutions by smooth solutions and the stability of this approximating procedure. And we concluded that this equation is everywhere ill-posed in the super-critical regime. Next, we considered the quintic beam equation on 3D torus. And we showed that this equation is almost sure global well-posed in certain super-critical regime. Lastly, we proved that the image measure of the Gaussian measure under the generalized BBM flow map satisfies a log-Sobolev type inequality with a little bit loss of integrability.