Split-step Finite-difference Time-domain Research Articles

A Novel Unconditionally Stable SS-FDTD Method With Isotropic Dispersion

A novel unconditionally stable split-step finite-difference time domain (SS-FDTD) method based on an isotropic dispersion finite-difference scheme is introduced. The proposed method reformed the weighting factor and scaling factor of the isotropic-dispersion FDTD for the SS-FDTD method with four substeps, which can generate nearly exact phase velocity for a single frequency. Compared with the conventional four-stage split-step FDTD method, the numerical dispersion error of the new method is significantly reduced even for a large stability factor. The new method is validated by numerical simulations.

Unified integro-differential equation for efficient dispersive FDTD simulations

PurposeThe purpose of this paper is to derive a unified formulation for incorporating different dispersive models into the explicit and implicit finite difference time domain (FDTD) simulations.Design/methodology/approachIn this paper, dispersive integro-differential equation (IDE) FDTD formulation is presented. The resultant IDE is written in the discrete time domain by applying the trapezoidal recursive convolution and central finite differences schemes. In addition, unconditionally stable implicit split-step (SS) FDTD implementation is also discussed.FindingsIt is found that the time step stability limit of the explicit IDE-FDTD formulation maintains the conventional Courant–Friedrichs–Lewy (CFL) constraint but with additional stability limits related to the dispersive model parameters. In addition, the CFL stability limit can be removed by incorporating the implicit SS scheme into the IDE-FDTD formulation, but this is traded for degradation in the accuracy of the formulation.Research limitations/implicationsThe stability of the explicit FDTD scheme is bounded not only by the CFL limit but also by additional condition related to the dispersive material parameters. In addition, it is observed that implicit JE-IDE FDTD implementation decreases as the time step exceeds the CFL limit.Practical implicationsBased on the presented formulation, a single dispersive FDTD code can be written for implementing different dispersive models such as Debye, Drude, Lorentz, critical point and the quadratic complex rational function.Originality/valueThe proposed formulation not only unifies the FDTD implementation of the frequently used dispersive models with the minimal storage requirements but also can be incorporated with the implicit SS scheme to remove the CFL time step stability constraint.

Unconditionally stable split-step finite difference time domain formulations for double-dispersive electromagnetic materials

Systematic split-step finite difference time domain (SS-FDTD) formulations, based on the general Lie–Trotter–Suzuki product formula, are presented for solving the time-dependent Maxwell equations in double-dispersive electromagnetic materials. The proposed formulations provide a unified tool for constructing a family of unconditionally stable algorithms such as the first order split-step FDTD (SS1-FDTD), the second order split-step FDTD (SS2-FDTD), and the second order alternating direction implicit FDTD (ADI-FDTD) schemes. The theoretical stability of the formulations is included and it has been demonstrated that the formulations are unconditionally stable by construction. Furthermore, the dispersion relation of the formulations is derived and it has been found that the proposed formulations are best suited for those applications where a high space resolution is needed. Two-dimensional (2-D) and 3-D numerical examples are included and it has been observed that the SS1-FDTD scheme is computationally more efficient than the ADI-FDTD counterpart, while maintaining approximately the same numerical accuracy. Moreover, the SS2-FDTD scheme allows using larger time step than the SS1-FDTD or ADI-FDTD and therefore necessitates less CPU time, while giving approximately the same numerical accuracy.

Unified Frequency-Dependent Split-Step FDTD Formulations Based on the Complex-Conjugate Pole-Residue Pairs Model

Unified and efficient frequency-dependent split-step finite difference time domain (SS-FDTD) formulations are presented. In the proposed formulations, the complex-conjugate pole-residue (CCPR) pairs model is adopted in the FDTD implementation of the domain's frequency-dependent relation with the minimal additional auxiliary variables. In addition, the formulations allow modeling different dispersive materials, such as Debye, Drude, and Lorentz, in the same manner. A numerical example is included to show the validity of the formulations.

Reinterpreting Four-Stage Split-Step FDTD Methods as Two-Stage Methods

In recent years, split-step finite-difference time-domain (SS-FDTD) methods have attracted much attention and many implementations have been developed to improve their accuracy and efficiency. In this communication the equivalence between some of these recently reported techniques and conventional schemes is shown. More specifically, we prove that there are four-stage split-step (4SS)-FDTD algorithms that can be reinterpreted as classic two-stage split-step (2SS)-FDTD schemes yielding the same local truncation errors and numerical dispersion relationships. The analytical results are validated by numerical simulations.

A Compact Unconditionally Stable Method for Time-Domain Maxwell's Equations

Higher order unconditionally stable methods are effective ways for simulating field behaviors of electromagnetic problems since they are free of Courant-Friedrich-Levy conditions. The development of accurate schemes with less computational expenditure is desirable. A compact fourth-order split-step unconditionally-stable finite-difference time-domain method (C4OSS-FDTD) is proposed in this paper. This method is based on a four-step splitting form in time which is constructed by symmetric operator and uniform splitting. The introduction of spatial compact operator can further improve its performance. Analyses of stability and numerical dispersion are carried out. Compared with noncompact counterpart, the proposed method has reduced computational expenditure while keeping the same level of accuracy. Comparisons with other compact unconditionally-stable methods are provided. Numerical dispersion and anisotropy errors are shown to be lower than those of previous compact unconditionally-stable methods.

A Novel Absorbing Boundary Condition for the 3-D Split-Step FDTD Method

In this letter, a novel absorbing boundary condition (ABC) is proposed for the 3-D split-step finite-difference time-domain (FDTD) method. Compared with the traditional Mur first-order ABC, the proposed ABC is simpler and requires less additional computation. Numerical simulation results show that the reflection error of the proposed ABC is much lower than that of the traditional Mur first-order ABC and its modifications. Further investigation shows that the accuracy of the proposed ABC is almost equivalent to the Mur second-order ABC.

Split-step finite-difference time-domain method with perfectly matched layers for efficient analysis of two-dimensional photonic crystals with anisotropic media

This Letter presents a split-step (SS) finite-difference time-domain (FDTD) method for the efficient analysis of two-dimensional (2-D) photonic crystals (PhCs) with anisotropic media. The proposed SS FDTD method is formulated with perfectly matched layer boundary conditions and caters for inhomogeneous anisotropic media. Furthermore, the proposed method is derived using the efficient SS1 splitting formulas with simpler right-hand sides that are more efficient and easier to implement. A 2-D PhC cavity with anisotropic media is used as an example to validate the efficiency of the proposed method.

A Split-Step FDTD Method for 3-D Maxwell's Equations in General Anisotropic Media

A split-step finite-difference time-domain (FDTD) method is presented for 3-D Maxwell's equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. Stability analysis of the proposed split-step FDTD method using the Fourier method is presented. The eigenvalues of the Fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the split-step FDTD method in general anisotropic media. Numerical results are presented to further validate the accuracy and stability of the proposed split-step FDTD method in general anisotropic media.

On the Accuracy of Three Different Six Stages SS-FDTD Methods for Modeling Narrow-Band Electromagnetic Applications

Recently, three different six stages split-step finite difference time domain (SS-FDTD) methods introduced for removing the FDTD Courant-Friedrichs-Levy (CFL) stability limit. Accuracy performance of these methods for modeling narrow-band electromagnetic applications is studied. Numerical example carried out in two-dimensional domain shows that when the CFL stability limit is exceeded, the accuracy of the classical FDTD scheme is maintained only with the splitting scheme that solves the field equations at each FDTD sub-stage alternatively between the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</i> directions.

Split-step finite difference time domain algorithm for double negative meta-materials

Unconditionally stable finite difference time domain (FDTD) formulation is presented for modelling electromagnetic wave propagation in open region double-negative meta-material domains. The formulation is based on incorporating the bilinear transformation technique and the stretched coordinate perfectly matched layer into the split-step FDTD algorithm. Numerical examples carried out in one- and two-dimensional domains show that the proposed formulation not only maintains the accuracy level of the conventional alternating direction implicit FDTD scheme but also reduces the required computational time by 21%.

Strang Split-Step FDTD Formulations for Modeling Dispersive Electromagnetic Applications

Unconditionally stable split-step finite-difference time-domain (SS-FDTD) formulations are presented for modeling dispersive electromagnetic applications. The formulations are based on incorporating the Strang split-step approach into the complex envelope FDTD (CE-FDTD) algorithm. A numerical example carried out in two-dimensional (2D) domain shows that the proposed formulations provide better accuracy than the CE locally one-dimensional SS-FDTD (CE-LOD-FDTD) counterpart with a considerable reduction in the CPU time requirement.

Unconditionally Stable Bilinear Transformation FDTD Algorithm for Modeling Double-Negative Meta-material Electromagnetic Problems

Accurate and unconditionally stable finite difference time domain (FDTD) algorithm is presented for modeling electromagnetic wave propagation in double-negative (DNG) meta-material domains. The proposed algorithm is based on incorporating the Bilinear transformation technique into the FDTD implementations of Maxwell’s equations. The stability of the proposed approach is studied by combining the von Neumann method with the Routh-Huwitz criterion and it has been observed that the proposed algorithm is free from the Courant-Friedrichs-Lewy (CFL) stability limit of the conventional FDTD scheme. Furthermore, the proposed algorithm is incorporated with the split-step FDTD scheme to model two-dimensional problems. Numerical examples carried out in one and two dimensional domains are included to show the validity of the proposed algorithm.

High‐order accurate FDTD method based on split‐step scheme for solving Maxwell's equations

AbstractA new split‐step finite difference time domain (SS‐FDTD) method with high‐order accuracy is presented, which is proven to be unconditionally stable and has four substeps. The numerical dispersion error and the numerical anisotropic error of the proposed method are reduced than the alternating direction implicit finite difference time domain method, the conventional SS‐FDTD method and the SS‐FDTD method based on the Strang‐splitting scheme. The proposed method has new splitting forms, which is different from the SS‐FDTD method based on the exponential evolution operator. At each time step, an important aspect is that the proposed method produces 33% reduction of the total number of arithmetic operators than the SS‐FDTD method based on the exponential evolution operator. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 562–565, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24100

Complex Envelope Six-Stages Split-Step PML Algorithm for Open Region Electromagnetic Problems

An implicit complex envelope split-step perfectly matched layer algorithm is presented for modeling open region finite difference time domain (FDTD) problems. The algorithm is based on splitting the FDTD time-step into six stages with an equal time increment of Delta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> /6 and it is suitable for band-limited electromagnetic applications. Numerical examples carried out in two dimensions show that the proposed algorithm presents a significant improvement over the other existing six-stages split-step FDTD algorithm for different Courant-Friedrichs-Lewy numbers and for different cell sizes per wavelength.

An Improved Implicit Split-Step Algorithm for Dispersive Band-Limited FDTD Applications

An improved unconditionally stable envelope finite difference time domain (FDTD) algorithm is presented for modeling open region dispersive band-limited electromagnetic applications. The algorithm is based on incorporating the split-step implicit FDTD scheme into the stretched coordinates perfectly matched layer absorbing boundary conditions. Numerical examples carried out in 2-D domains are included to show the validity of the proposed formulations.

High-order accurate split-step FDTD method for solution of Maxwell's equations

The split-step finite difference time domain (SS-FDTD) method characterised by unconditional stability is becoming an important numerical method in computational electromagnetics. Proposed is a new high-order accurate unconditionally stable SS-FDTD method, which is derived from the exponential evolution operator. Compared with the conventional SS-FDTD method, the numerical dispersion of the new method is greatly reduced.