| There are two main topics in this dissertation. The first one presents a general framework of developing stable spectral methods for solving partial differential equations defined on arbitrary domains using tetrahedral elements with unstructured grids. The second topic is a study of the finite difference method for numerical simulations in electromagnetics.; In the first topic, the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the Jacobi polynomials to problems beyond the line. This allows for the identification of nodal sets suitable for polynomial interpolations within the tetrahedron. We then formulate energy-stable schemes based on the nodal sets for solving advection, advection-diffusion and linear symmetric hyperbolic equations. In addition, factorization methods for efficient computations of derivatives on the nodal sets are also provided. The performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain, confirming the expected spectral accuracy and stability.; In the second topic, a detailed stability analysis of the Yee scheme is presented first. The main result from the analysis shows that the use of the “magic time step” in the Yee scheme may result in an unstable numerical computation. The second part of this topic is devoted to a discussion of modeling dielectric material interfaces in the finite difference method. Traditionally, such problems are solved by using a technique which is a combination of staircase formulation and averaged material parameters. However, one often finds that the accuracy of the results from this method is very poor. This is mainly due to the staircase formulation. To overcome this unfortunate situation, a new method is presented. The new technique uses one-sided difference operator and dielectric interface boundary conditions and avoids the side effects from staircase interfaces. A comparison between these methods directly by numerical computations is provided and it is shown that the new method is superior to the traditional ones. |
