A Highly Accurate FDTD Model for Simulating Lorentz Dielectric Dispersion

A highly accurate and numerically stable model of Lorentz dielectric dispersion for the finite-difference time-domain (FDTD) method is presented. The coefficients of the proposed model are optimally derived based on the Maclaurin series expansion (MSE) method and it is shown that the model is much better than the other four reported models in implementing the Lorentz dielectric dispersion with error of relative permittivity several orders lower. The model's stability and performance are also analyzed when it is incorporated into the practical second- and fourth-order accurate FDTD algorithms for an exemplified Lorentz medium. Interestingly, we find that all the mentioned models show nearly the same performance in the second-order algorithm due to its large intrinsic numerical dispersion and the superiority of the proposed MSE model begins to be manifested in the higher-order, say, fourth-order FDTD algorithms as implied by the governing numerical dispersion equations.      

碰撞等离子体的高阶FDTD算法

给出了电磁波在均匀、碰撞等离子体中传播的四阶时间和四阶空间FDTD算法.该算法比Yee氏FDTD算法每一个网格每一维增加一个存储单元,与常规的二阶等离子体FDTD算法相同.由于采用四阶时间和四阶空间近似,因此该算法能有效地减小数字色散误差,其频带宽度比二阶算法的频带宽度更宽.为了验证该高阶算法的正确性,对均匀、碰撞等离子体平板的电磁波反射系数进行了计算,并与解析结果、二阶FDTD计算结果进行了比较,证明了该算法的高效和精确.     

Relative accuracy of several finite-difference time-domain methodsin two and three dimensions

A comparison of the accuracy of several orthogonal-grid finite-difference-time-domain (FDTD) schemes is made in both two and three-dimensions. The relative accuracy is determined from the dispersion error associated with each algorithm and the number of floating-point operations required to obtain a desired accuracy level. In general, in both 2-D and 3-D, fourth-order algorithms are more efficient than second-order schemes in terms of minimizing the number of computations for a given accuracy level. In 2-D, a second-order approach proposed by Z. Chen et al. (1991) is much more accurate than the scheme of K.S. Yee (1966) for a given amount of computation, and can be as efficient as fourth-order algorithms. In 3-D, Yee's algorithm is slightly more efficient than the approach of Chen et al. in terms of operations, but much more efficient in terms of memory requirements     

A hybrid fourth-order FDTD utilizing a second-order FDTD subgrid

A hybrid method utilizing the second-order accurate in time and fourth-order accurate in space FDTD (2, 4) coupled with the standard second-order accurate both in time and space FDTD (2, 2) on a subgrid is presented. The accuracy of the method is tested by computing the S-parameters of two monopoles mounted on a ground plane and it is found to be very satisfactory. Significant computational savings both in memory and time are accomplished by using this hybrid method     

A low-dispersion 3-D second-order in time fourth-order in space FDTD scheme (M3d/sub 24/)

A second-order in time fourth-order in space modified finite-difference time-domain (FDTD) scheme for three dimensional electromagnetic problems "M3d/sub 24/" is presented. The algorithm enables the numerical phase error to be minimized, so that it leads to high accuracy with low resolution grids. The advantage of this method is demonstrated by considering the long distance propagation of the wave radiated from a time harmonic elementary dipole using a low resolution grid, and comparing the results with other FDTD schemes.     

An Alternative Algorithm for Both Narrowband and Wideband Lorentzian Dispersive Materials Modeling in the Finite-Difference Time-Domain Method

In this study, an alternative algorithm is proposed for modeling narrowband and wideband Lorentzian dispersive materials using the finite-difference time-domain (FDTD) method. Previous algorithms for modeling narrowband and wideband Lorentzian dispersive materials using the FDTD method have been based on a recursive convolution technique. They present two different and independent algorithms for the modeling of the narrowband and wideband Lorentzian dispersive materials, known as the narrowband and wideband Lorentzian recursive convolution algorithms, respectively. The proposed alternative algorithm may be used as a general algorithm for both narrowband and wideband Lorentzian dispersive materials modeling with the FDTD method. The second-order motion equation for the Lorentzian materials is employed as an auxilary differential equation. The proposed auxiliary differential-equation-based algorithm can also be applied to solve the borderline case dispersive electromagnetic problems in the FDTD method. In contrast, the narrowband and wideband Lorentzian recursive convolution algorithms cannot be used for the borderline case. A rectangular cavity, which is partially filled with narrowband and wideband Lorentzian dispersive materials, is presented as a numerical example. The time response of the electric field z component is used to validate and compare the results     

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.     

A higher order (2,4) scheme for reducing dispersion in FDTDalgorithm

A finite-difference time-domain (FDTD) scheme with second-order accuracy in time and fourth-order in space is discussed for the solution of Maxwell's equations in the time domain. Compared with the standard Yee (1966) FDTD algorithm, the higher order scheme reduces the numerical dispersion and anisotropy and has improved stability. Dispersion analysis indicates that the frequency band in which the higher order scheme yields an accurate solution is widened on the same grid, this means a larger space increment can be chosen for the same excitation. Numerical results show the applications of the scheme in modeling wide-band electromagnetic phenomena on a coarse grid     

A higher order FDTD method for EM propagation in a collisionlesscold plasma

A fourth-order in time and space, finite-difference time-domain (FDTD) scheme is presented for radio-wave propagation in a lossless cold plasma. As with previously reported fourth-order schemes, the methodology is founded on the principle that correction derivatives (i.e., three derivatives in time) can be converted into vector spatial derivatives. From the error analysis and phase-velocity data, it is argued that this approach will significantly minimize the dispersion errors while still maintaining minimal memory requirements. This claim is also supported by data obtained from FDTD simulations. Using a one-dimensional plasma slab problem as the test case, we show that the bandwidth and dynamic range associated with this fourth-order scheme are significantly improved with respect to its second-order counterpart. The impact of other error mechanisms, namely material boundary-related errors, is also discussed     

A modified FDTD (2, 4) scheme for modeling electrically largestructures with high-phase accuracy

A new fourth-order finite-difference time-domain (FDTD) scheme has been developed that exhibits extremely low-phase errors at low-grid resolutions compared to the conventional FDTD scheme. Moreover, this new scheme is capable of combining with the standard Yee (1966) scheme to produce a stable hybrid algorithm. The problem of wave propagation through a building is simulated using this new hybrid algorithm to demonstrate the large savings in computing resources it could afford. With this new development, the FDTD method can now be used to successfully model structures that are thousands of wavelengths large, using the present day computer technology     

Higher-order FDTD(2,4) scheme for accurate simulations in lossy dielectrics

A finite-difference time-domain (FDTD)(2,4) scheme with second-order accuracy in time and fourth-order accuracy in space for the precise solution of Maxwell's equations in lossy dielectrics is presented. Compared with the ordinary FDTD method the novel technique reduces lattice reflection errors, increases the overall accuracy and provides significant computational savings. Numerical results for a waveguide problem indicate the efficiency and robustness of the proposed formulation.     

Analysis of curved and angled surfaces on a Cartesian mesh using anovel finite-difference time-domain algorithm

The widely accepted finite-difference time-domain algorithm, based on a Cartesian mesh, is unable to rigorously model the curved surfaces which arise in many engineering applications, while more rigorous solution algorithms are inevitably considerably more computationally intensive. A nonintensive, but still rigorous, alternative to this approach has been to incorporate a priori knowledge of the behavior of the fields (their asymptotic static field solutions) into the FDTD algorithm. Unfortunately, until now, this method has often resulted in instability. In this contribution an algorithm (denoted `SFDTD' for second-order finite difference time domain) is presented which uses the static field solution technique to accurately characterize curved and angled metallic boundaries. A hitherto unpublished stability theory for this algorithm, relying on principles of energy conservation, is described and it is found that for the first time a priori knowledge of the field distribution can be incorporated into the algorithm with no possibility of instability. The accuracy of the SFDTD algorithm is compared to that of the standard FDTD method by means of two test structures for which analytic results are available     

A Versatile Split-Field 1-D Propagator for Perfect FDTD Plane Wave Injection

A new staggered field design and formulation for the one-dimensional propagator of the total-field/scattered-field source implementation in finite-difference time domain (FDTD) scattering simulations are presented. The new equations are based on split-field Maxwell's equations and the resulting technique extends the functionality of the multipoint auxiliary propagator to sourcing FDTD lattices hosting extended-stencil high-order algorithms. This technique virtually eliminates numerical dispersion, field location and polarization mismatches between propagator and main grid. The resulting machine accuracy-level leakage error from implementing this technique is confirmed for the standard low and high-order FDTD schemes as well as the M24 high-order algorithm. Normalized field leakage for all three algorithm implementations outside the total-field region was measured at below - 295 dB.     

Nonstandard FDTD Method for Wideband Analysis

The nonstandard (NS) FDTD algorithm can compute electromagnetic propagation with very high accuracy on a coarse grid, but only for monochromatic or narrow-band signals. We have developed a wideband (W) NS-FDTD algorithm that overcomes this limitation. In NS-FDTD special finite difference operators are used to make the numerical dispersion isotropic, which is then corrected by a frequency-dependent factor. In WNS-FDTD the numerical dispersion is modeled as frequency-dependent electrical permittivity and magnetic permeability, and the Yee algorithm is augmented by correction terms in the time domain. We demonstrate the high accuracy of WNS-FDTD in example problems, and show that it gives better results than both the standard (S) FDTD and the FDTD(2,4) algorithms.      

FDTD Schemes With Minimal Numerical Dispersion

A novel formulation of hybrid finite-difference time-domain (FDTD) methods is presented. Significant reduction of numerical dispersion is achieved by the proposed FDTD methods that combine the second-order and higher-order finite-differences. Also, the proposed FDTD methods exhibit significantly higher solution accuracy than the accuracy of standard FDTD schemes as a result of partial mutual cancellation of numerical errors provided by the developed FDTD update procedure. The residual numerical error of the phase velocity remains low even for sampling of a few points per wavelength. Also, the FDTD schemes based on the proposed approach are faster and more accurate than the corresponding purely higher-order FDTD schemes with the same mesh. Test examples are provided for validation purposes.      

The unconditionally stable pseudospectral time-domain (PSTD) method

This letter presents a new time-domain method for Maxwell's equations, in which the unconditionally stable techniques, the alternating direction implicit (ADI) and the split-step (SS) schemes, are developed for the pseudospectral time-domain (PSTD) algorithm to maintain stability while achieving higher accuracy and efficiency over the FDTD method. The multidomain strategy is employed to allow for a flexible treatment of internal inhomogeneities. Numerical results demonstrate the unconditional stability and the second-order accuracy for both ADI- and SS-PSTD algorithms.     

Filtered-X second-order Volterra adaptive algorithms

The authors propose filtered-X second-order Volterra adaptive algorithms based on a multichannel structure application to active noise control. The developed algorithms can be used as alternatives in the case where the standard filtered-X LMS algorithm does not perform well     

Numerical and experimental corroboration of an FDTD thin-slot modelfor slots near corners of shielding enclosures

Simple design maxims to restrict slot dimensions in enclosure designs below a half-wave length are not always adequate for minimizing electromagnetic interference (EMI). Complex interactions between cavity modes, sources, and slots can result in appreciable radiation through nonresonant length slots. The finite-difference time domain (FDTD) method can be employed to pursue these issues with adequate modeling of thin slots. Subcellular FDTD algorithms for modeling thin slots in conductors have previously been developed. One algorithm based on a quasistatic approximation has been shown to agree well with experimental results for thin slots in planes. This FDTD thin-slot algorithm is compared herein with two-dimensional (2-D) moment method results for thin slots near corners and plane wave excitation. FDTD simulations are also compared with measurements for slots near an edge of a cavity with an internal source     

A Fourth-Order Runge–Kutta in the Interaction Picture Method for Simulating Supercontinuum Generation in Optical Fibers

An efficient algorithm, which exhibits a fourth-order global accuracy, for the numerical solution of the normal and generalized nonlinear Schrodinger equations is presented. It has applications for studies of nonlinear pulse propagation and spectral broadening in optical fibers. Simulation of supercontinuum generation processes, in particular, places high demands on numerical accuracy, which makes efficient high-order schemes attractive. The algorithm that is presented here is an adaptation for use in the nonlinear optics field of the fourth-order Runge-Kutta in the Interaction Picture (RK4IP) method, which was originally developed for studies on Bose-Einstein condensates. The performance of the RK4IP method is validated and compared to a number of conventional methods by modeling both the propagation of a second-order soliton and the generation of supercontinuum radiation. It exhibits the expected global fourth-order accuracy for both problems studied and is the most accurate and efficient of the methods tested.     

Second-order split-step envelope PML algorithm for 2D FDTD simulations

Unconditionally stable second-order split-step (SS) envelope perfectly matched layer (PML) formulations are presented for truncating finite-difference time-domain (FDTD) grids. The proposed method is based on the second-order SS-FDTD algorithm. Numerical examples carried out in two-dimensional domains are included to show the validity of the proposed formulations.