Chun-Hao Teng

Stable Spectral Methods on Tetrahedral Elements

A framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented. Although most of the developments are of a general nature, an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron. In the first part 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 interpolation within the tetrahedron and, subsequently, for the formulation of stable spectral schemes on such unstructured nodal sets. The second part of this work is devoted to a discussion of weakly imposed boundary conditions, and energy-stable schemes are formulated for a wide class of problems, exemplified by advection problems, advection-diffusion problems, and linear symmetric hyperbolic systems. Finally, in the third part, issues related to computational efficiency and implementation of the schemes are discussed. The spectral accuracy of the approximation is confirmed through an example, and factorization methods for the efficient computation of derivatives on the general nodal sets within the d-simplex are developed, ensuring that the proposed schemes are competitive with tensor-product-based methods. In this last part we also show that the advective operator results in an

A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell's Equations

{\cal O}(n^{-2})

β-Adrenergic modulation of arrhythmogenesis and identification of targeted sites of antiarrhythmic therapy in Timothy (LQT8) syndrome: a theoretical study

restriction on the time-step, similar to that of spectral collocation methods employing a tensor-product-based approximation. The performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain, confirming the expected accuracy and stability.

Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation.

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme.

A Multidomain Pseudospectral Mode Solver for Optical Waveguide Analysis

Timothy syndrome (TS) is a malignant form of congenital long QT syndrome with a mode of arrhythmia onset often triggered by enhanced sympathetic tone. We sought to explore mechanisms by which beta-adrenergic stimulation (BAS) modulates arrhythmogenesis and to identify potential targeted sites of antiarrhythmic therapy in TS. Using a dynamic Luo-Rudy ventricular myocyte model incorporated with detailed intracellular Ca(2+) cycling, along with its one-dimensional multicellular strand, we simulated various clinical scenarios of TS, with stepwise increase in the percentage of G406R Ca(v)1.2 channels from 0 to 11.5 and 23%, and to 38.5 and 77%, respectively, for heterozygous and homozygous states of TS1 and TS2. Progressive prolongation of action potential duration (APD) and QT interval, accompanied by amplification of transmural dispersion of repolarization, steepening of APD restitution, induction of delayed afterdepolariztions (DADs), and both DAD and phase 3 early afterdepolariztion-mediated triggered activities, correlated well with the extent of G406R Ca(v)1.2 channel mutation. BAS amplified transmural dispersion of repolarization, steepened APD restitution, and facilitated inducibility of DAD-mediated triggered activity. Systematic analysis of intracellular Ca(2+) cycling revealed that sarcoplasmic reticulum Ca(2+) ATPase (uptake current) played an essential role in BAS-induced facilitation of DAD-mediated triggered activity and, in addition to L-type calcium current, it could be an effective site of antiarrhythmic therapy under the influence of BAS. Thus G406R Ca(v)1.2 channel mutation confers not only a trigger, but also a substrate for lethal ventricular arrhythmias, which can be exaggerated by BAS. It is suggested that, besides beta-adrenergic blockers and L-type calcium current channel blockers, an agent aimed at reduction of sarcoplasmic reticulum Ca(2+) ATPase uptake current may provide additional antiarrhythmic effect in patients with TS.

Simulations of Dielectric and Plasmonic Waveguide-Coupled Ring Resonators Using the Legendre Pseudospectral Time-Domain Method

In this study, we report the investigation of both near- and far-field electromagnetic characteristics of two-dimensional silver nanorod arrays embedded in anodic aluminum oxide with the use of a high-accuracy three-dimensional Legendre pseudospectral time-domain scheme. The simulated far-field scattering spectra agree with the experimental observations. We show that enhanced electric field is created between adjacent nanorods and, most importantly, far-field scattered light wave is mainly contributed from surface magnetic field, instead of the surface enhanced electric field. The identified near-field to far-field connection produces an important implication in the development of efficient surface-enhanced Raman scattering substrates.

A Pseudospectral Penalty Scheme for 2D Isotropic Elastic Wave Computations

We propose a pseudospectral mode solver for optical waveguide mode analysis formulated by the frequency-domain Maxwell equations. Special attention is paid upon identifying the required boundary operator for the formulation and the relationships between the derived operator and the physical boundary conditions. These theoretical results are adopted into a Legendre pseudospectral multidomain computational framework to compute the propagation characteristics of optical waveguides. Numerical experiments are conducted, and the expected spectral convergence of the scheme is observed for smooth problems and for problems having field jumps at material interfaces. For dielectric waveguides with sharp corners, the spectral convergence is lost due to the singular nature of fields at the corner. Nevertheless, compared with other methods, the present formulation remains as an efficient approach to obtain waveguide modes.

High-Accuracy Waveguide Leaky-Mode Analysis Using a Multidomain Pseudospectral Frequency-Domain Method Incorporated With Stretched Coordinate PML

The accurate Legendre pseudospectral time-domain (PSTD) method is applied in this paper to the study of optical behaviors of 2-D dielectric and plasmonic waveguide-coupled ring resonators. Based on its inherent multidomain scheme with curvilinear-quadrilateral subdomain partitioning, the PSTD method possesses superior advantage in fulfilling field continuity conditions across material interfaces, which provides better numerical handling upon circular or even more complicated ring structures. Accordingly, investigations of several plasmonic nanoring resonators of different shapes with surface plasmon polariton (SPP) waves propagating on the metal/dielectric/metal waveguides are made with the aids of a well-fitted Drude-Lorentz material-dispersion model through analyzing their characteristics in the near-infrared and visible parts of the light spectrum. Especially, the racetrack-shaped ring is carefully designed with optimized track length to avoid the over- or under-coupling problem between the ring and input waveguides and, thus, achieve excellent spectral performance.

Improved Analysis of Rectangular Dielectric Waveguides Based on a Legendre Pseudospectral Penalty Scheme

Abstract In this paper, we present a pseudospectral scheme for solving 2D elastic wave equations. We start by analyzing boundary operators leading to the well-posedness of the problem. In addition, equivalent characteristic boundary conditions of common physical boundary conditions are discussed. These theoretical results are further employed to construct a Legendre pseudospectral penalty scheme based on a tensor product formulation for approximating waves on a general curvilinear quadrilateral domain. A stability analysis of the scheme is conducted for the case where a straight-sided quadrilateral element is used. The analysis shows that, by properly setting the penalty parameters, the scheme is stable at the semi-discrete level. Numerical experiments for testing the performance of the scheme are conducted, and the expected p- and h-convergence patterns are observed. Moreover, the numerical computations also show that the scheme is time stable, which makes the scheme suitable for long time simulations.

Application of Pseudospectral Methods to Optical Waveguide Mode Solvers

A recently developed pseudospectral frequency-domain (PSFD) method employing a penalty scheme is further incorporated with stretched coordinate perfectly matched layers (PMLs) in order to solve waveguide leaky modes. Several standard leaky-mode slab and circular waveguides, including the W-type and M-type ones, are examined first by solving their complex effective indexes and field profiles, showing that the PSFD method achieves computational accuracy on the order of 10-15. Then, this high-accuracy solver is used to analyze waveguide structures with more complicated geometries, such as the leaky six-air-hole fiber, for which self convergence of accuracy in calculated effective indexes is demonstrated to be on the orders of 10-14 , and two rectilineal waveguides with sharp corners, i.e., the rib waveguide and the photonic wire. Comparison of the PSFD method obtained results for these more complicated waveguide structures with those from other methods in the literature is presented and discussed.