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Controlling and understanding the propagation of optical pulses through dispersive media forms the basis for optical communication, medical imaging, and other modern technological advances. Integral to this control and understanding is the ability to describe the transients which occur immediately after the onset of a signal. This thesis examines the transients of such a system when a unit step function is applied. The electromagnetic field is described by an integral resulting from Maxwell’s Equations. It was previously believed that optical precursors, a specific transient effect, existed only for only a few optical cycles and contributed only small magnitudes to the field. The main results of this thesis show that the transients arising from this integral are entirely precursors and that they may exist on longer time scales and contribute larger magnitudes to the field. The experimental detection of precursors has previously been recognized only through success comparison to the transient field resulting from an application of the method of steepest descent to that field integral. For any parameter regime where steepest descents may be applied, this work gives iterative methods to determine saddle points which are both more accurate than the accepted results and to extend into regimes where the current theory has failed. Furthermore, asymptotic formulae have been derived for regions where previous attempts at steepest descent have failed. Theory is also presented which evaluates the applicability of steepest descents in the represention of precursor behavior for any set of parameters. Lastly, the existence of other theoretical models for precursor behavior who may operate beyond the reach of steepest descent is validated through successful comparisons of the transient prediction of those methods to the steepest descent based results of this work. / Dissertation

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 102-112).

Thesis (MScIng)--University of Stellenbosch, 2007. / Bibliography.

Thesis (M.S.)--Ohio State University, 2006. / Includes bibliographical references (leaves 61-63). Available online via OhioLINK's ETD Center

Chan Kit Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 90-93). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Model Elliptic Boundary-Value Problems --- p.2 / Chapter 1.2 --- Applications of the Model Boundary-Value Problem --- p.4 / Chapter 1.2.1 --- Curl-Curl Formulation --- p.4 / Chapter 1.2.2 --- Vector Potential Formulation --- p.6 / Chapter 1.2.3 --- Darwin Model and Quasistatic Model --- p.7 / Chapter 1.3 --- Spurious Solutions --- p.8 / Chapter 2 --- Finite Element Formulation --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Weak Formulation --- p.14 / Chapter 2.2.1 --- Galerkin Method --- p.17 / Chapter 2.2.2 --- The Rayleigh-Ritz Method --- p.19 / Chapter 2.3 --- H1(Ω) Conforming Finite Element Method --- p.23 / Chapter 2.3.1 --- The Dirichlet Problem --- p.24 / Chapter 2.3.2 --- The Neumann Problem --- p.27 / Chapter 3 --- Numerical Implementations --- p.29 / Chapter 3.1 --- Introduction --- p.29 / Chapter 3.2 --- Implementation of Boundary Conditions --- p.32 / Chapter 3.3 --- Numerical Integration Formula --- p.39 / Chapter 3.4 --- Discrete L2-norms --- p.40 / Chapter 3.5 --- Solution of Linear System of Equations --- p.42 / Chapter 3.6 --- Automatic Mesh Generation --- p.43 / Chapter 3.6.1 --- The Cubic Domain Ω --- p.44 / Chapter 3.6.2 --- The Spherical Shell Domain Ωs --- p.44 / Chapter 4 --- Numerical Experiments --- p.50 / Chapter 4.1 --- Numerical Experiments for Dirichlet Problem --- p.50 / Chapter 4.1.1 --- Original Formulation --- p.50 / Chapter 4.1.2 --- Experiments --- p.52 / Chapter 4.1.3 --- Penalty Factor Effect --- p.56 / Chapter 4.2 --- Numerical Experiment for Neumann Problem --- p.61 / Chapter 4.2.1 --- Original Formulation --- p.61 / Chapter 4.2.2 --- Experiments --- p.62 / Chapter 4.2.3 --- Penalty Factor Effect --- p.66 / Chapter 4.2.4 --- Comparison with the Dirichlet Problem --- p.70 / Chapter 4.3 --- Numerical Experiment of Dirichlet Problem with Boundary Condition E = E --- p.71 / Chapter 4.3.1 --- Original Formulation --- p.71 / Chapter 4.3.2 --- Experiments --- p.73 / Chapter 4.3.3 --- Penalty Factor Effect --- p.76 / Chapter 4.4 --- Numerical Experiment on Spherical Shell Domain --- p.81 / Chapter 4.4.1 --- The Spherical Shell Domain --- p.81 / Chapter 4.4.2 --- Dirichlet Problem --- p.82 / Chapter 4.5 --- Some Numerical Phenomena --- p.86 / Chapter 4.5.1 --- GMRES Convergence Accelerator --- p.86 / Chapter 4.5.2 --- Sparsity Improvement --- p.88 / Bibliography --- p.90 / List of Tables --- p.94

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering in the Lightning and EMC Research Group School of Electrical and Information Engineering September 2015 / The work presented contributes to research in lightning protection simulations and focuses on approximating the Heidler function with an analytical integral and hence a frequency domain representation. The integral of lightning current models is required in the analysis of lightning events including the induced effects and frequency analyses of lightning strikes. Previous work in this area has produced very specific forms of the Heidler function that are used to represent lightning current waveshapes. This work however focuses on a generic solution with parameters that can be modified to produce any lightning current waveshape that is required. In the research presented, such an approximation is obtained. This function has an analytical solution to the integral and hence can be completely represented in the frequency domain. This allows for a true representation of Maxwell’s equations for Electromagnetic (EM) fields and for an analytical frequency domain analysis. It has parameters that can be changed to obtain different waveshapes (10/350, 0.25/100, etc.). The characteristics of the approximation are compared with those of the Heidler function to ascertain whether or not the function is applicable for use with the lightning protection standard (IEC 62305-1). It is shown that the approximation does represent the same characteristics as those of the Heidler function and hence can be used in IEC 62305-1 standardised applications. This represents a valuable contribution to engineers working in the field of lightning protection, specifically simulation models. / MT2017

Chung Tsz Shun Eric. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 102-104). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Applications of Maxwell's equations --- p.1 / Chapter 1.2 --- Introduction to Maxwell's equations --- p.2 / Chapter 1.3 --- Historical outline of numerical methods --- p.4 / Chapter 1.4 --- A new approach --- p.5 / Chapter 2 --- Mathematical Backgrounds --- p.7 / Chapter 2.1 --- Sobolev spaces --- p.7 / Chapter 2.2 --- Tools from functional analysis --- p.8 / Chapter 3 --- Discretization of Vector Fields --- p.10 / Chapter 3.1 --- Domain triangulation --- p.10 / Chapter 3.2 --- Mesh dependent norms --- p.11 / Chapter 3.3 --- Discrete circulation operators --- p.13 / Chapter 3.4 --- Discrete flux operators --- p.20 / Chapter 4 --- Spatial Discretization of the Maxwell's Equations --- p.23 / Chapter 4.1 --- Derivation --- p.23 / Chapter 4.2 --- Consistency theory --- p.29 / Chapter 4.3 --- Convergence theory --- p.33 / Chapter 4.3.1 --- Polyhedral domain --- p.33 / Chapter 4.3.2 --- Rectangular domain --- p.38 / Chapter 5 --- Fully Discretization of the Maxwell's Equations --- p.63 / Chapter 5.1 --- Derivation --- p.63 / Chapter 5.2 --- Consistency theory --- p.65 / Chapter 5.3 --- Convergence theory --- p.69 / Chapter 5.3.1 --- Polyhedral domain --- p.69 / Chapter 5.3.2 --- Rectangular domain --- p.77 / Chapter 6 --- Numerical Tests --- p.97 / Chapter 6.1 --- Convergence test --- p.97 / Chapter 6.2 --- Electromagnetic scattering --- p.99 / Bibliography --- p.102

Maxwell’s equations are a set of partial differential equations that describe the classic electromagnetic problems, electrodynamics etc. Effective numerical methods are derived to solve the equations in the past decades, and continued to be of great interest to be developed to its completion. In this thesis, we introduce and propose numerical methods using staggered meshes that deal with both two dimensional and three dimensional space problem in polygonal and general curved domains. / Finite difference method, finite volume method, spectral method and staggered discontinuous Galerkin method are discussed in the thesis. A forth order finite difference method using Taylor expansion technic is proposed. The integral form of the original Maxwell’s equations give rise to methods based on more general domain. For the finite volume method, covolume methods both on the cyclic polygon elements and noncyclic polygon elements are derived. To derive a higher order accurate method, staggered discontinuous Galerkin method based on the same domain decomposition present in the finite volume method use Nedelec elements is derived in two dimensional space, and spectral method using nodal high-order method operate on a general domain in 3D with flexible domain geometry is introduced. Numerical results are shown to show the performance oft he above mentioned approximation methods in 2D case. / 麥克斯韋方程組是一組描述經典電磁問題，電磁力學的偏微分方程。在過去數十年，行之有效的偏微分方程數值解已經被推導出並用於求解該方程，該問題現在仍然吸引著學者極大的興趣，並日臻完善。在這篇論文中，我們介紹並提出一些運用曲域交錯網格數值方法在二維和三維的多面體和更一般幾何體處理麥克斯韋方程組問題。 / 本論文對有限差分法，有限體積法，光譜法和交錯間斷有限元方法進行了討論。利用泰勒展開式這一方法推導出一個二維的四階有限差分方法。基於原來的麥克斯韋方程組的積分形式所得到的數值方法更適用於更普遍的域。對於有限體積法，對循環多邊形元素和非環狀多邊形元素的有限體積方法都將被導出。為了得到一個更高階準確的方法，基於有限體積法中使用的域分解方法，使用Nedelec元素，推導了二維空間的高階有限元方法。基於頂點高階數值方法的光譜法對於三維一般定義域的幾何形態更為靈活適用。在二維的定義域中，數值模擬結果驗證上述數值方法的精確性。 / Jian, Fangqiong. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 62-65). / Abstracts also in Chinese. / Title from PDF title page (viewed on 07, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 204-208).