Nonstandard Finite Differences Research Articles

High-Accuracy FDTD Solution of the Absorbing Wave Equation, and Conducting Maxwell's Equations Based on a Nonstandard Finite-Difference Model

We previously introduced high-accuracy finite-difference time-domain (FDTD) algorithms based on nonstandard finite differences (NSFD) to solve the nonabsorbing wave equation and the nonconducting Maxwell equations. We now extend our methodology to the absorbing wave equation and the conducting Maxwell equations. We first derive an exact NSFD model of the one-dimensional wave equation, and extend it to construct high-accuracy FDTD algorithms to solve the absorbing wave equation, and the conducting Maxwell's Equations in two and three dimensions. For grid spacing h, and wavelength /spl lambda/, the NSFD solution error is /spl epsiv//spl sim/(h//spl lambda/)/sup 6/ compared with (h//spl lambda/)/sup 2/ for ordinary FDTD algorithms using second-order central finite-differences. This high accuracy is achieved not by using higher-order finite differences but by exploiting the analytical properties of the decaying-harmonic solution basis functions. Besides higher accuracy, in the NSFD algorithms the maximum time step can be somewhat longer than for the ordinary second-order FDTD algorithms.

Investigations of nonstandard, Mickens‐type, finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

AbstractIt is well known that standard finite‐difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (𝒪(1/r) or 𝒪(log r)) behavior of the solution near the origin (r = 0). New nonstandard finite‐difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these “Mickens‐type” finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co‐ordinates are investigated. The numerical results generated by the Mickens‐type schemes are compared favorably with solutions obtained from standard finite‐difference schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 380–398, 2003.

Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm

Propagation properties of optical pulses in a two-dimensional photonic crystal with a straight waveguide structure imbedded were examined using a high accuracy finite-difference time-domain (FDTD) algorithm based on nonstandard finite differences. A tunable and significantly large group velocity dispersion was found even for photonic crystal structures as small as 10 unit cells long. Detailed calculations indicated that a very small photonic crystal with an imbedded waveguide can be used to control pulse dispersion, i.e., a just 25 μm long photonic crystal with waveguide can compress a 1% up-chirped pulse to the Fourier transform limit. Further, our FDTD calculations showed excellent agreement with the prediction of photonic band calculations on infinite structures.

High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications

We previously described a high-accuracy version of the Yee algorithm that uses second-order nonstandard finite differences (NSFDs) and demonstrated its accuracy numerically. We now prove that at fixed frequency and grid spacing h, the leading error term is O(h/sup 6/) versus O(h/sup 2/) for the ordinary Yee algorithm with standard finite differences (SFDs). We numerically verify the superior accuracy of the NSFD algorithm by simulating near-field Mie scattering on a coarse grid and comparing with the SFD one and with analytical solutions. We present an updated stability analysis and show that the maximum time step for the NSFD algorithm is 20% longer than the SFD time step in two dimensions, and 36% longer in three dimensions. Finally, parameters that were previously given numerically are now analytically defined.

Simulation of light propagation in two-dimensional photonic crystals with a point defect by a high-accuracy finite-difference time-domain method

A high-accuracy finite-difference time-domain method based on what are called nonstandard finite differences was used to simulate optical propagation in a two-dimensional photonic crystal with a point defect. We used a photonic crystal consisting of a triangular lattice of air columns embedded in a high-refractive index medium. We found that the transmittance spectrum has four peaks in the photonic band-gap region, and that these peaks correspond to the resonant energies of light localized at the point defect. For a point defect consisting of an air hole with a radius smaller than that of the air holes of the photonic crystal, these peaks shift to higher energy. The peak shift of the resonant mode that is associated with the electric field concentrated about the center of the point defect is larger than the peak shift of the other modes.

High Accuracy Finite-Difference Time-Domain Simulation of Light Propagation in Photonic Crystals.

We have introduced a high accuracy version of the Yee algorithm based on nonstandard finite differences for which the error is e∼(h/λ)2 compared with e∼(h/λ)6 for the ordinary one. This high accuracy allows the use of coarse numerical grids, which greatly reduces the computational cost. Realistic two-dimensional calculations can run on a personal computer, while three-dimensional ones can run on a fast workstation. Using this high accuracy algorithm, we developed computational models and methods to simulate optical propagation in photonic crystals, and to compute their transmission spectra, and bandgap structures. In addition, methodologies were developed to accurately represent arbitrary lattice structures on coarse grids.

Finite Difference Time Domain Computation of Light Scattering by Multiple Colloidal Particles

A finite difference time domain (FDTD) algorithm with nonstandard finite differences is used to compute light scattering by colloidal particles. The objective of this research is to simulate scattering for cases where multiparticle scattering is significant and where particle shape is arbitrary and not limited to regular geometries commonly illustrated in mathematical solutions such as the Mie theory. In this paper scattering characteristics computed using the FDTD method are compared with known analytical solutions for a number of cases for which a two-dimensional treatment is sufficient. These include scattering by a single infinite cylinder, a hollow cylinder, and two parallel cylinders. The numerical results were found to agree well with the analytical solutions. Results are also given for cases where multiparticle scattering is significant and analytical solutions are extremely difficult if not impossible to obtain.

Correction To "a High-accuracy Realization Of The Yee Algorithm Using Non-standard Finite Differences"

Correction To "a High-accuracy Realization Of The Yee Algorithm Using Non-standard Finite Differences"

Generalized nonstandard finite differences and physical applications

Nonstandard finite differences can be used to construct exact algorithms to solve some differential equations of physical interest such as the wave equation and Schrödinger’s equation. Even where exact algorithms do not exist, nonstandard finite differences can greatly improve the accuracy of low-order finite-difference algorithms with a computational cost low compared to higher-order schemes or finer gridding. While nonstandard finite differences have been applied successfully to a variety of one-dimensional problems, they cannot be directly extended to higher dimensions without modification. In this article we generalize the nonstandard finite-difference methodology to two and three dimensions, give example algorithms, and discuss practical applications. © 1998 American Institute of Physics.

A high-accuracy realization of the Yee algorithm using non-standard finite differences

New nonstandard second-order finite differences (FD's) are introduced, which when substituted into the Yee algorithm, reduce the solution error by a factor of 10/sup -4/ on a coarse computational grid. Using /spl lambda//h (grid spacings per wavelength)=8, one achieves the same accuracy as the standard Yee algorithm does at /spl lambda//h=1140. In addition, greater algorithmic stability allows a reduction in the number of iterations needed to solve a problem.

High accuracy solution of Maxwell’s equations using nonstandard finite differences

We introduce a new finite-difference time-domain algorithm to directly solve Maxwell’s equations based on nonstandard finite differences. This algorithm is some 10,000 times more accurate than the standard one on a coarse grid. Although computational load per grid point is greater, it is more than offset by a large reduction in the total number of grid points needed to solve a given problem. In addition, algorithm stability is greater, so that the number of iterations needed is also reduced. While optimum performance is achieved at a fixed frequency, the accuracy is still higher than that of the standard algorithm over moderate bandwidths. The algorithm is implemented in Fortran 90 and can easily model spatially variant media and irregular boundaries. By displaying one or more fields per wave period we obtain on-line movielike visualizations of the electromagnetic fields while the computation is running. © 1997 American Institute of Physics.