Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisors: Kenneth S. Ball and David G. Bogard. Vita. Includes bibliographical references.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Also available online.

Thesis (M.S.)--Oregon State University, 2008. / Printout. Includes bibliographical references (leaf 24). Also available on the World Wide Web.

The focus of the research was on numerical errors committed in foundation phase mathematics. It therefore explored: (1) numerical errors learners in foundation phase mathematics encounter (2) relationships underlying numerical errors and (3) the implementable strategies suitable for understanding numerical error analysis in foundation phase mathematics (Grade 3). From 375 learners who formed the population of the study in the primary schools (16 in total), the researcher selected by means of a simple random sample technique 80 learners as the sample size, which constituted 10% of the population as response rate. On the basis of the research questions and informed by positivist paradigm, a quantitative approach was used by means of tables, graphs and percentages to address the research questions. A Likert scale was used with four categories of responses ranging from (A) Agree, (S A) Strongly Agree, (D) Disagree and (S D) Strongly Disagree. The results revealed that: (1) the underlying numerical errors that learners encounter, include the inability to count backwards and forwards, number sequencing, mathematical signs, problem solving and word sums (2) there was a relationship between committing errors and a) copying numbers b) confusion of mathematical signs or operational signs c) reading numbers which contained more than one digit (3) It was also revealed that teachers needed frequent professional training for development; topics need to change and lastly government needs to involve teachers at ground roots level prior to policy changes on how to implement strategies with regards to numerical errors in the foundational phase. It is recommended that attention be paid to the use of language and word sums in order to improve cognition processes in foundation phase mathematics. Moreover, it recommends that learners are to be assisted time and again when reading or copying their work, so that they could have fewer errors in foundation phase mathematics. Additionally it recommends that teachers be trained on how to implement strategies of numerical error analysis in foundation phase mathematics. Furthermore, teachers can use tests to identify learners who could be at risk of developing mathematical difficulties in the foundation phase.

Research on interfacial phenomena has a long history, which has attracted tremendous interest in recent years. One of the most successful tools is the phase-field approach. As phase-field models usually involve very complex dynamics and it is nontrivial to obtain analytical solutions, numerical methods have played an important role in various simulations. This thesis is mainly devoted to developing accurate, efficient and robust numerical methods and the related numerical analysis for three representative phase-field models, namely the Allen-Cahn equation, the Cahn-Hilliard equation and the thin film models. The first part of this thesis is mainly concentrated on the stability analysis for these three models, with particular attention to the Allen-Chan equation. We have established three stability criterion, i.e., nonlinear energy stability, L∞-stability and L2-stability. As shared by most phase-field models, one of the intrinsic properties of the Allen- Cahn and the Cahn-Hilliard equations is that they satisfy a nonlinear stability re- lationship, usually expressed as a time-decreasing free energy functional. We have studied several stabilized temporal discretization for both the Allen-Cahn and the Cahn-Hilliard equations so that the relevant nonlinear energy stability can be pre- served. The corresponding temporal discretization schemes are linear and are of second-order accuracy. We also apply multi-step implicit-explicit methods to ap- proximate the Allen-Cahn equation. We demonstrate that by suitably choosing the parameters in multi-step implicit-explicit methods the nonlinear energy stability can be preserved. Apart from studying the energy stability for the Allen-Cahn equation, we also establish the numerical maximum principle for some fully discretized schemes. We further extend our analysis technique to the fractional-in-space Allen-Cahn equation. A more general Allen-Cahn-type equation with a nonlinear degenerate mobility and a logarithmic free energy is also considered. The third stability under investigation is the L2-stability. We prove that the con- tinuum Allen-Cahn equation satisfies a uniform Lp-stability. Furthermore, we show that both semi-discretized Fourier Galerkin and Fourier collocation methods can in- herit this stability for p = 2, i.e., L2-stability. Based on the established L2-stability, we accomplish the spectral convergence estimate for the Fourier Galerkin methods. We adopt the second-order Strang splitting schemes in the temporal direction with Fourier collocation methods to demonstrate the uniform L2-stability in the fully dis- cretized scheme. Another contribution of this thesis is to propose a p-adaptive spectral deferred correction methods for the long time simulations for all three models. We develop a high-order accurate and energy stable scheme to simulate the phase-field models by combining the semi-implicit spectral deferred correction method and first-order stabilized semi-implicit schemes. It is found that the accuracy improvement may affect the overall energy stability. To compromise the accuracy and stability, a local p- adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. Numerical results demonstrate the high effectiveness of our proposed numerical strategy. Keywords: Phase-field models, Allen-Cahn equations, Cahn-Hilliard equations, thin film models, nonlinear energy stability, maximum principle, L2-stability, adaptive simulations, stabilized semi-implicit schemes, finite difference, Fourier spectral meth- ods, spectral deferred correction methods, convex splitting

We have a conjecture that “the precision of computed solutions for systems of linear equations is more closely related with componentwise condition number c(A) than normwise condition number κ(A). We conducted simulation experiments to verify this conjecture. A statistical tool, Hotelling-Williams T-Test is employed to check if difference between correlations is significant. Simulation results suggest that our conjecture is true for most of the well-known methods and matrix sizes. Keywords: condition numbers, simulation, correlation coefficients, Hotelling-Williams T-Test

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第14836号 / 工博第3133号 / 新制||工||1469(附属図書館) / 27242 / UT51-2009-F478 / 京都大学大学院工学研究科マイクロエンジニアリング専攻 / (主査)教授 小寺 秀俊, 教授 北條 正樹, 教授 田畑 修 / 学位規則第4条第1項該当

This thesis introduces and analyses a numerical method for solving time-dependent partial differential equations (PDEs) on surfaces. This method is based on the closest point method, and solves the surface PDE by solving a suitably chosen equation in a band surrounding the surface. As it uses an implicit closest point representation of the surface, the method has the advantages of being simple to implement for very general surfaces, and amenable to discretization with a broad class of numerical schemes. The method proposed in this work introduces a new equation in the embedding space, which satisfies a key consistency property with the surface PDE. Rather than alternating between explicit time-steps and re-extensions of the surface function as in the original closest point method, we investigate an alternative approach, in which a single equation can be solved throughout the embedding space, without separate extension steps. This is achieved by creating a modified embedding equation with a penalty term, which enforces a constraint on the solution. The resulting equation admits a method of lines discretization, and can therefore be discretized with implicit or explicit time-stepping schemes, and analysed with standard techniques. The method can be formulated in a straightforward way for a large class of problems, including equations featuring variable coefficients, higher-order terms or nonlinearities. The effectiveness of the method is demonstrated with a range of examples, drawing from applications involving curvature-dependent diffusion and systems of reaction-diffusion equations, as well as equations arising in PDE-based image processing on surfaces.

We consider the finite difference method applied to a class of financial problems. Specifically, we investigate the properties of the Du Fort and Frankel finite difference scheme and experiment with adaptations of the scheme to improve on its consistency properties. The Du Fort and Frankel finite difference scheme is applied to a number of problems that frequently occur in finance. We specifically investigate problems associated with jumps, discontinuous behavior, free boundary problems and multi dimensionality. In each case we consider adaptations to the Du Fort and Frankel scheme in order to produce reliable results. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

South Africa is a water scarce country where a large number of people still are not provided with adequate water. This project will help to manage water resources in a sustainable manner that will benefit the country as a whole. This study concentrated on the behaviour of pipe materials with leak openings under different pressure conditions. The stress distribution throughout a pipe is known due to the pressures within the pipe i.e. the longitudinal and circumferential stresses as well as the working pressure of the pipe, however when a leak opening such as a hole or crack appears in the pipe the stress distribution around these openings change. It is the effect of this stress distribution that this study addressed. The effect of stresses around these leak openings may provide some knowledge as to how and when the pipes will fail completely as well as help in explaining the leakage exponents within pipes. Using the method of Finite Element Analysis (FEA) the project aimed to find the relationship between the pressure in the pipe and the behaviour of the leak openings for different leak types. The different leak openings investigated were: a circular hole, a longitudinal crack and a circumferential crack. This study used finite element analysis to understand what happens to a pipe when pressure is applied within the pipe and the pipe has a leak opening e.g. a small hole, a longitudinal crack or a circumferential crack in it. The materials studied were uPVC, steel, cast iron and fibre-cement. The main conclusions drawn from this study were that when a pipe has a circular hole in it the leakage exponent does not differ from the theoretical value of 0.5. With the longitudinal cracks it was found that regardless of whether there are longitudinal stresses in the pipe or not the leakage exponent is the same: and these exponents vary significantly from the theoretical value of 0.5 and can be in the order of 0.87, 0.82, 0.75, 0.55 for a 60 mm long crack in an uPVC, steel, cast iron and fibre-cement pipe respectively. With the circumferential cracks however, there were significant differences in exponents due to the longitudinal and lack of longitudinal stress. For the case when the pipe has the longitudinal stresses the exponents were significantly larger than 0.5. The exponents were in the order of 1.15, 1.02, 0.95, and 0.64 for a 150 mm long crack of uPVC, steel, cast iron and fibre-cement respectively. If there is no longitudinal stress then the exponents were smaller than 0.5 and tended to close the crack along the circumference. The exponents were 0.45, 0.47, 0.46, and 0.49 for a 150 mm crack of uPVC, steel, cast iron and fibre-cement respectively. / Prof. J.E. van Zyl