Optimized operator-splitting methods in numerical integration of Maxwell's equations

Optimized operator splitting methods for numerical integration of the time domain Maxwell's equations in computational electromagnetics (CEM) are proposed for the first time. The methods are based on splitting the time domain evolution operator of Maxwell's equations into suboperators, and corresponding time coefficients are obtained by reducing the norm of truncation terms to a minimum. The general high-order staggered finite difference is introduced for discretizing the three-dimensional curl operator in the spatial domain. The detail of the schemes and explicit iterated formulas are also included. Furthermore, new high-order Padé approximations are adopted to improve the efficiency of the proposed methods. Theoretical proof of the stability is also included. Numerical results are presented to demonstrate the effectiveness and efficiency of the schemes. It is found that the optimized schemes with coarse discretized grid and large Courant-Friedrichs-Lewy (CFL) number can obtain satisfactory numerical results, which in turn proves to be a promising method, with advantages of high accuracy, low computational resources and facility of large domain and long-time simulation. In addition, due to the generality, our optimized schemes can be extended to other science and engineering areas directly. © 2012 Z. X. Huang et al.published_or_final_versio

Stable coupling of the Yee scheme with a linear current model

This work analyzes the stability of the Yee scheme for non stationary Maxwell's equations coupled with a linear current model. We show that the usual procedure may yield unstable scheme for physical situations that correspond to strongly magnetized plasmas in X-mode (TE) polarization. We propose to use first order clustered discretization of the vectorial product that gives back a stable coupling. We validate the schemes on some test cases representative of direct numerical simulations of X-mode in a magnetic fusion plasma

High Order Energy - Conserved Splitting FDTD Methods for Maxwell's Equations

Computation of Maxwell's equations has been playing an important role in many applications, such as the radio-frequencies, microwave antennas, aircraft radar, integrated optical circuits, wireless engineering and materials, etc. It is of particular importance to develop numerical methods to solve the equations effectively and accurately. During the propagation of electromagnetic waves in lossless media without sources, the energies keep constant for all time, which explains the physical feature of electromagnetic energy conservations in long term behaviors. Preserving the invariance of energies is an important issue for efficient numerical schemes for solving Maxwell's equations. In my thesis, we first develop and analyze the spatial fourth order energy-conserved splitting FDTD scheme for Maxwell's equations in two dimensions. For each time stage, while the spatial fourth-order difference operators are used to approximate the spatial derivatives on strict interior nodes, the important feature is that on the near boundary nodes, we propose a new type of fourth-order boundary difference operators to approximate the derivatives for ensuring energy conservative. The proposed EC-S-FDTD-(2,4) scheme is proved to be energy-conserved, unconditionally stable and of fourth order convergence in space. Secondly, we develop and analyze a new time fourth order EC-S-FDTD scheme. At each stage, we construct a time fourth-order scheme for each-stage splitting equations by converting the third-order correctional temporal derivatives to the spatial third-order differential terms approximated further by the three central difference combination operators. The developed EC-S-FDTD-(4,4) scheme preserves energies in the discrete form and in the discrete variation forms and has both time and spatial fourth-order convergence and super-convergence. Thirdly, for the three dimensional Maxwell's equations, we develop high order energy-conserved splitting FDTD scheme by combining the symplectic splitting with the spatial high order near boundary difference operators and interior difference operators. Theoretical analyses including energy conservations, unconditional stability, error estimates and super-convergence are established for the three dimensional problems. Finally, an efficient Euler-based S-FDTD scheme is developed and analyzed to solve a very important application of Maxwell's equations in Cole-Cole dispersive medium. Numerical experiments are presented in all four parts to confirm our theoretical results

ADI schemes for the time integration of Maxwell equations

This thesis is concerned with the analysis and construction of alternating direction implicit (ADI) splitting schemes for the time integration of linear isotropic Maxwell equations on cuboids. The work is organized in two major parts. The first part deals with time discrete approximations to exponentially stable Maxwell equations. By means of a divergence cleaning technique and artificial damping, we obtain an ADI scheme with approximations that also decay exponentially in time. The decay rate is here uniform with respect to the time discretization. One of the main ingredients in the proof of the decay behavior is an observability estimate for the numerical approximations. This inequality is obtained by means of a discrete multiplier technique. We also provide a rigorous error analysis for the uniformly exponentially stable ADI scheme, yielding convergence of order one in a space similar to $H^{-1}$. The error result makes only assumptions on the initial data and the model parameters. In the second part, we analyze time discrete approximations to linear isotropic Maxwell equations on a heterogeneous cuboid. In this setting, the domain consists of two different homogeneous subcuboids. The Maxwell equations are here integrated in time by means of the Peaceman-Rachford ADI splitting scheme which is well-known in literature. The main result provides a rigorous error bound of order 3/2 in $L^2$ for the numerical approximations. It is significant that the final error statement involves conditions only on the initial data and model parameters, but not on the solution. To achieve this result, we establish a detailed regularity analysis for the considered Maxwell system

Towards efficient three-dimensional wide-angle beam propagation methods and theoretical study of nanostructures for enhanced performance of photonic devices

In this dissertation, we have proposed a novel class of approximants, the so-called modified Padé approximant operators for the wide-angle beam propagation method (WA-BPM). Such new operators not only allow a more accurate approximation to the true Helmholtz equation than the conventional operators, but also give evanescent modes the desired damping. We have also demonstrated the usefulness of these new operators for the solution of time-domain beam propagation problems. We have shown this both for a wideband method, which can take reflections into account, and for a split-step method for the modeling of ultrashort unidirectional pulses. The resulting approaches achieve high-order accuracy not only in space but also in time. In addition, we have proposed an adaptation of the recently introduced complex Jacobi iterative (CJI) method for the solution of wide-angle beam propagation problems. The resulting CJI-WA-BPM is very competitive for demanding problems. For large 3D waveguide problems with refractive index profiles varying in the propagation direction, the CJI method can speed-up beam propagation up to 4 times compared to other state-of-the-art methods. For practical problems, the CJI-WA-BPM is found to be very useful to simulate a big component such as an arrayed waveguide grating (AWG) in the silicon-on-insulator platform, which our group is looking at. Apart from WA beam propagation problems for uniform waveguide structures, we have developed novel Padé approximate solutions for wave propagation in graded-index metamaterials. The resulting method offers a very promising tool for such demanding problems. On the other hand, we have carried out the study of improved performance of optical devices such as label-free optical biosensors, light-emitting diodes and solar cells by means of numerical and analytical methods. We have proposed a solution for enhanced sensitivity of a silicon-on-insulator surface plasmon interference biosensor which had been previously proposed in our group. The resulting sensitivity has been enhanced up to 5 times. Furthermore, we have developed an improved model to investigate the influence of isolated metallic nanoparticles on light emission properties of light-emitting diodes. The resulting model compares very well to experimental results. Finally, we have proposed the usefulness of core-shell nanostructures as nanoantennas to enhance light absorption of thin-film amorphous silicon solar cells. An increased absorption up to 33 % has theoretically been demonstrated

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement