Dissertation (PhD)--University of Stellenbosch, 2006. / Please refer to full text for abstract.

This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.

A two-dimensional numerical model has been developed to simulate wave-current induced nearshore circulation patterns in beaches and surf zones. The wave model is based on the parabolic wave equation for mild slope beaches. The parabolic equation method has been chosen because it is a viable means of predicting the characteristics of surface waves in slowly varying domains and in its present form dissipation and wave breaking are also included. The two dimensional parabolic mild slope equation was discretised and solved in a fully implicit manner, so stability did not create a major problem. This wave model was then embedded into the existing numerical model DIVAST. The sediment transport formulae from Van Rijn was used to calculate the nearshore sediment transport rate.

We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

This thesis investigates inversion techniques used to determine the geoacoustic properties of a shallow-water waveguide. The data used were obtained in the Shallow Water '06 Modal Mapping Experiment in which four buoys drifted over a system of subbottom channels. The method used was perturbative inversion using modal eigenvalues as input parameters, which were found using an autoregressive spectral estimator. This work investigates the differences between a "channel" region and a "no channel" region based on an inferred stratigraphic model. Inversions were performed on data from a single buoy both at individual frequencies and multiple frequencies simultaneously. Since the use of multiple frequencies and a certain set of constraints proved to be an effective method of inversion, the method was applied to data from the other three buoys as well. It is shown that the "channel" and "no channel" regions have significantly different sound speed profiles. / by Rebecca Weeks. / Thesis (M.S.C.S.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

Sound propagation in a waveguide is greatly dependent on the acoustic properties of the boundaries. The effect of these properties can be described by a bottom reflection coefficient RB, and surface reflection coefficient RS. Two methods for estimating reflection coefficients are used in this research. The first, the ratio method, is based on the variations of the Green's function with depth utilizing the ratio of the wavenumber spectra at two depths. The second, the pole method, is based on the wavenumbers of the modal peaks in the spectrum at a particular depth. A method to invert for sound speed and density is also examined. Estimates of RB and RS based on synthetic data by the ratio method were very close to their predicted values, especially for higher frequencies and longer apertures. The pole method returned less precise estimates though with longer apertures, the estimates were better. Using experimental data, results of the pole method as well a geoacoustic inversion technique based on them were mixed. The ratio method was used to estimate RS based on the actual data and returned results close to the predicted phase of p. / by Alexander Conrad. / Vita. / Thesis (M.S.C.S.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

La thèse porte sur l’analyse d’erreur a posteriori pour la résolution numérique de l’équation linéaire des ondes , discrétisée en temps par le schéma de Newmark et en espace par la méthode des éléments finis. Nous adoptons un choix particulier de paramètres pour le schéma de Newmark, notamment β = 1/4, γ = 1/2, qui assure que la méthode est conservative en énergie et d’ordre deux en temps. L’estimation d’erreur a posteriori, d’un ordre optimal en temps et en espace, est élaborée à partir de la discrétisation complète. L’erreur est mesurée dans une norme qui découle naturellement de la physique: H1 en espace et Linf en temps. Nous proposons d’abord un estimateur dit «à 3 points» qui fait intervenir la solution discrète en 3 points successifs du temps à chaque pas de temps. Cet estimateur fait appel à une approximation du Laplacien de la solution discrète qui doit être calculée à chaque pas de temps, en résolvant un problème auxiliaire d'éléments finis. Nous proposons ensuite un estimateur d’erreur alternatif qui permet d’éviter ces calculs supplémentaires: l’estimateur dit «à 5 points» puisqu’il met en jeu le schéma des différences finies d’ordre 4, qui fait intervenir la solution discrète en 5 points successifs du temps à chaque pas de temps. Nous démontrons que nos estimateurs en temps sont d’ordre optimal pour des solutions suffisamment lisses, sur des maillages quasi-uniformes en espace et uniformes en temps, en supposant que les conditions initiales soient discrétisées à l’aide de la projection elliptique. La trouvaille la plus intéressante de cette analyse est le rôle capitale de cette discrétisation : des discrétisations standards pour les conditions initiales, telles que l’interpolation nodale, peuvent être néfastes pour les estimateurs d’erreur en détruisant leur ordre de convergence, bien qu’elles fournissent des solutions numériques tout à fait acceptables. Des expériences numériques prouvent que nos estimateurs d’erreur sont d’ordre optimal en temps comme en espace, même dans les situations non couvertes par la théorie. En outre, notre analyse a posteriori s’étend au schéma de Newmark d’ordre deux plus général (γ = 1/2). Nous présentons des comparaisons numériques entre notre estimateur à 3 points généralisé et l’estimateur sur des grilles décalées, proposé par Georgoulis et al. Finalement, nous implémentons un algorithme adaptatif en temps et en espace basé sur notre estimateur d’erreur a posteriori à 3 points. Nous concluons par des expériences numériques qui montrent l’efficacité de l’algorithme adaptatif et révèlent l’importance de l’interpolation appropriée de la solution numérique d’un maillage à un autre, surtout vis à vis de l’optimalité de l’estimation d’erreur en temps. / This thesis focuses on the a posteriori error analysis for the linear second-order wave equation discretized by the second order Newmark scheme in time and the finite element method in space. We adopt the particular choice for the parameters in the Newmark scheme, namely β = 1/4, γ = 1/2, since it provides a conservative method with respect to the energy norm. We derive a posteriori error estimates of optimal order in time and space for the fully discrete wave equation. The error is measured in a physically natural norm: H1 in space, Linf in time. Numerical experiments demonstrate that our error estimators are of optimal order in space and time. The resulting estimator in time is referred to as the 3-point estimator since it contains the discrete solution at 3 points in time. The 3-point time error estimator contains the Laplacian of the discrete solution which should be computed via auxiliary finite element problems at each time step. We propose an alternative time error estimator that avoids these additional computations. The resulting estimator is referred to as the 5-point estimator since it contains the fourth order finite differences in time and thus involves the discrete solution at 5 points in time at each time step. We prove that our time estimators are of optimal order at least on sufficiently smooth solutions, quasi-uniform meshes in space and uniform meshes in time. The most interesting finding of this analysis is the crucial importance of the way in which the initial conditions are discretized: a straightforward discretization, such as the nodal interpolation, may ruin the error estimators while providing quite acceptable numerical solution. We also extend the a posteriori error analysis to the general second order Newmark scheme (γ = 1/2) and present numerical comparasion between the general 3-point time error estimator and the staggered grid error estimator proposed by Georgoulis et al. In addition, using obtained a posteriori error bounds, we implement an efficient adaptive algorithm in space and time. We conclude with numerical experiments that show that the manner of interpolation of the numerical solution from one mesh to another plays an important role for optimal behavior of the time error estimator and thus of the whole adaptive algorithm.

In this thesis, we consider four different stochastic partial differential equations. Firstly, we study stochastic Helmholtz equation driven by an additive white noise, in a bounded convex domain with smooth boundary of Rd (d = 2, 3). And then with the help of the perfectly matched layers technique, we also consider the stochastic scattering problem of Helmholtz type. The second part of this thesis is to investigate the time harmonic case for stochastic Maxwell's equations driven by an color noise in a simple medium, and then we expand the results to the stochastic Maxwell's equations in case of dispersive media in Rd (d = 2, 3). Thirdly, we study stochastic parabolic partial differential equation driven by space-time color noise, where the domain O is a bounded domain in R2 with boundary ∂O of class C2+alpha for 0 < alpha < 1/2. In the last part, we discuss the stochastic wave equation (SWE) driven by nonlinear noise in 1D case, where the noise 626x6t W(x, t) is the space-time mixed second-order derivative of the Brownian sheet. / Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. / One of the goals of this thesis is to derive error estimates for numerical solutions of the above four kinds SPDEs. The difficulty in the error analysis in finite element methods and general numerical approximations for a SPDE is the lack of regularity of its solution. To overcome such a difficulty, we follow the approach of [4] by first discretizing the noise and then applying standard finite element methods and discontinuous Galerkin methods to the stochastic Helmholtz equation and Maxwell equations with discretized noise; standard finite element method to the stochastic parabolic equation with discretized color noise; Galerkin method to the stochastic wave equation with discretized white noise, and we obtain error estimates are comparable to the error estimates of finite difference schemes. / We shall focus on some SPDEs where randomness only affects the right-hand sides of the equations. To solve the four types of SPDEs using, for example, the Monte Carlo method, one needs many solvers for the deterministic problem with multiple right-hand sides. We present several efficient deterministic solvers such as flexible CG method and block flexible GMRES method, which are absolutely essential in computing statistical quantities. / Zhang, Kai. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 144-155). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*} To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz. Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

The most common tool for solving spacetime problems using finite elements is based on semidiscretization: discretizing in space by a finite element method and then advancing in time by a numerical scheme. Contrary to this standard procedure, in this dissertation we consider formulations where time is another coordinate of the domain. Therefore, spacetime problems can be studied as boundary value problems, where initial conditions are considered as part of the spacetime boundary conditions. When seeking solutions to these problems, it is natural to ask what are the correct spaces of functions to choose, to obtain wellposedness. This motivates the study of an abstract theory for unbounded partial differential operators associated with a general boundary value problem on a bounded domain. A framework for choosing the spaces is introduced, and conditions for the solvability of weak formulations are provided. We apply this framework to study wave problems on tents and to study wellposed discontinuous Petrov-Galerkin (DPG) formulations for the Schrödinger and wave equations. Several numerical issues are also discussed.