| This thesis contains two topics on high-order accurate methods for solving Maxwell's equations. The first topic is the application of high-order accurate methods to the modeling/designs of photonic crystals. The second topic is the development of thin layer approximations and implementation in discontinuous Galerkin method. Each topic consists of two parts as described below.;The first part, part I of the first topic, demonstrates the behavior and sensitivity of the frozen mode phenomenon in finite structures with anisotropic materials, including both magnetic materials and non-normal incidence. A discontinuous Galerkin method is used for solving Maxwell's equations in the time domain. The existence of the frozen mode phenomena is confirmed and the impact of finiteness and perturbations of periodicity is studied.;In the second part, part II of the second topic, PDE constrained nonlinear optimization techniques are implemented to optimized the design of electromagnetic crystals for the frozen mode phenomenon. We investigate both of gyrotropic photonic crystals and degenerate band edge crystals as well as the more complex case of the oblique incidence. We extend the investigation to the three-dimensional case to identify the first three-dimensional crystal exhibiting frozen mode behavior.;The third part, part I of the second topic, studies high-order accurate thin layer approximations for metal backed coatings in the time-domain. Isotropic materials and tangentially-oriented anisotropic materials are considered in one and two dimensions. The implementation of these models are discussed in the context of discontinuous Galerkin methods which are particularly well-suited for these approximations. The range of validity, accuracy, and stability of the resulting schemes is discussed through one- and two-dimensional examples.;The last part, part II of the second topic, also studies high-order accurate thin layer approximations, but for transmission layers. The thin layer formulation of 3rd order and 5th order for curvilinear transmission layers are derived. Also, similar to the previous part, computational results on the range of validity, accuracy, and stability is demonstrated on one- and two-dimensional cases.;This thesis is based on the following 4 papers: (1) S. Chun and J. S. Hesthaven, Modeling of the frozen mode phenomenon and its Sensitivity using Discontinuous Galerkin methods, commun. comput. phys., 2, 2007, pp. 611--639. (2) S. Chun and J. S. Hesthaven, PDE constrained optimization and design of frozen mode crystals, commun. comput. phys., 3, 2008, pp. 878--898. (3) S. Chun and J. S. Hesthaven, High-order accurate thin layer approximation for time-domain electromagnetics, Part I: General metal backed coatings, in preparation. (4) S. Chun, H. Haddar and J. S. Hesthaven, High-order accurate thin layer approximation for time-domain electromagnetics, Part II: Interface conditions, in preparation. |
