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A New Form of the Numerical Dispersion Relation of ADI-FDTD for the 3D Maxwell's Equations
Abstract
In this paper, a new form of the numerical dispersion relation of the alternating direction implicit finite difference time domain (ADI-FDTD) method for the 3D Maxwell equations is derived from the equivalent form of ADI-FDTD. It is shown that this new form is simpler than the original one and easy to be used to make further analysis. By comparison with the numerical dispersion relations of the Crank-Nicolson (CN) FDTD scheme and the FDTD scheme it is shown that the numerical dispersion relation of ADI-FDTD is much more closer to that of CN-FDTD than to that of FDTD. The relation between the new form and the original form of the numerical dispersion relation of ADI-FDTD is also given that they are equivalent to each other.
... To obtain the numerical dispersion relationship, a monochromatic wave with angular frequency ω is assumed as u n = ue jωn∆t , thus (7) can be expressed in the following way [29]. ...
In this paper, the di®erence scheme of the alternating- direction-implicit finite-difference time-domain (ADI-FDTD) method is replaced by the quasi isotropic (QI) spatial difference scheme to improve its numerical dispersion characteristics. The unconditional stability advantage of QI-ADI-FDTD is analytically proven and numerically verified. The numerical dispersion of the novel method can be dramatically reduced by choosing proper weighting factor. An example is simulated to demonstrate the accuracy and effciency of the proposed method.
Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.
We present an analytical study of the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method for solving time-varying Maxwell's equations and compare its accuracy with that of the Crank-Nicolson (CN) and Yee FDTD schemes. The closed form of the truncation error is obtained for two and three dimensions. The dependence of the truncation error on the square of the time step multiplied by the spatial derivatives of the fields is found to be a unique feature of the ADI-FDTD scheme. We illustrate the limitation on accuracy imposed by these truncation error terms by simulating a simple parallel-plate structure excited by a low-frequency voltage source. Excellent agreement is obtained between field data computed with the implicit CN scheme using time steps greatly exceeding the Courant limit and field data computed with the explicit Yee scheme. Such results are not obtained with ADI-FDTD.
In this paper, an unconditionally stable three-dimensional (3-D)
finite-difference time-method (FDTD) is presented where the time step
used is no longer restricted by stability but by accuracy. The principle
of the alternating direction implicit (ADI) technique that has been used
in formulating an unconditionally stable two-dimensional FDTD is
applied. Unlike the conventional ADI algorithms, however, the
alternation is performed in respect to mixed coordinates rather than to
each respective coordinate direction, Consequently, only two
alternations in solution marching are required in the 3-D formulations.
Theoretical proof of the unconditional stability is shown and numerical
results are presented to demonstrate the effectiveness and efficiency of
the method. It is found that the number of iterations with the proposed
FDTD can be at least four times less than that with the conventional
FDTD at the same level of accuracy
This paper presents a comprehensive analysis of numerical
dispersion of the recently developed unconditionally stable
three-dimensional finite-difference time-domain (FDTD) method where the
alternating-direction-implicit technique is applied. The dispersion
relation is derived analytically and the effects of spatial and temporal
steps on the numerical dispersion are investigated. It is found that the
unconditionally stable FDTD scheme has advantages over the conventional
FDTD of the Yee's scheme in modeling structures of fine geometry where a
graded mesh is required. The unconditionally stable FDTD allows the use
of a large time step in a region of fine meshes while maintaining
numerical dispersion errors smaller than those associated with the
region of coarse meshes
The characteristics of the waves guided along a plane [I] P. S. Epstein, “On the possibility of electromagnetic surface waves, ” Proc. Nat’l dcad. Sciences, vol. 40, pp. 1158-1165, Deinterface which separates a semi-infinite region of free cember 1954. space from that of a magnetoionic medium are investi- [2] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer, ” Proc. IEEE, vol. 51, pp. 317gated for the case in which the static magnetic field is 332, February 1963. oriented perpendicular to the plane interface. It is [3] &I. A. Gintsburg, “Surface waves on the boundary of a plasma in a magnetic field, ” Rasprost. Radwvoln i Ionosf., Trudy found that surface waves exist only when w,<wp and NIZMIRAN L’SSR, no. 17(27), pp. 208-215, 1960. that also only for angular frequencies which lie bet\\-een [4] S. R. Seshadri and A. Hessel, “Radiation from a source near a plane interface between an isotropic and a gyrotropic dielectric,” we and 1/42 times the upper hybrid resonant frequency. Canad. J. Phys., vol. 42, pp. 2153-2172, November 1964. The surface waves propagate with a phase velocity [5] G. H. Owpang and S. R. Seshadri, “Guided waves propagating along the magnetostatic field at a plane boundary of a semiwhich is always less than the velocity of electromagnetic infinite magnetoionic medium, ” IEEE Trans. on Miomave waves in free space. The attenuation rates normal to the Tbory and Techniques, vol. MTT-14, pp. 136144, March 1966. [6] S. R. Seshadri and T. T. \Vu, “Radiation condition for a maginterface of the surface wave fields in both the media are netoionic medium. ” to be Dublished. examined. Kumerical results of the surface wave characteristics are given for one typical case.
This extensively revised and expanded third edition of the Artech House bestseller, Computational Electrodynamics: The Finite-Difference Time-Domain Method, offers the most up-to-date and definitive resource on this critical method for solving Maxwell’s equations. Material new to the third edition includes:
• Advanced techniques for PSTD;
• Advanced techniques for unconditional numerical stability;
• Provably stable, hybrid FDTD-FE techniques;
• Hardware acceleration for personal computers;
• Convolutional PML ABCs;
• Four-level, two-electron atomic systems yielding optical gain;
• Photonic crystals, second-order nonlinear optical materials, plasmonics, and biophotonics;
• Emerging FDTD techniques for global ULF and ELF geophysical models of the Earth.
This single resource provides complete guidance on FDTD techniques and applications, from basic concepts, to the current state-of-the-art. It enables professionals to more efficiently and effectively design and analyze key electronics and photonics technologies, including wireless communications devices, high-speed digital and microwave circuits, and integrated optics. It is also ideal for senior-year and graduate-level university courses in computational electrodynamics.
This letter presents a numerical dispersion relation for the two-dimensional (2-D) finite-difference time-domain method based on the alternating-direction implicit time-marching scheme (2-D ADI-FDTD). The proposed analytical relation for 2-D ADI-FDTD is compared with those relations in the previous works. Through numerical tests, the dispersion equation of this work was shown as correct one for 2-D ADI-FDTD.
The numerical dispersion property of the two-dimensional
alternating-direction implicit finite-difference time-domain (2D ADI
FDTD) method is studied. First, we notice that the original 2D ADI FDTD
method can be divided into two sub-ADI FDTD methods: either the
x-directional 2D ADI FDTD method or the y-directional 2D ADI FDTD
method; and secondly, the numerical dispersion relations are derived for
both the ADI FDTD methods. Finally, the numerical dispersion errors
caused by the two ADI FDTD methods are investigated. Numerical results
indicate that the numerical dispersion error of the ADI FDTD methods
depends highly on the selected time step and the shape and mesh
resolution of the unit cell. It is also found that, to ensure the
numerical dispersion error within certain accuracy, the maximum time
steps allowed to be used in the two ADI FDTD methods are different and
they can be numerically determined
A new, comprehensive CAD-oriented modeling methodology for single
and coupled interconnects on an Si-SiO2 substrate is
presented. The modeling technique uses a modified quasi-static spectral
domain electromagnetic analysis which takes into account the skin effect
in the semiconducting substrate. Equivalent-circuit models with only
ideal lumped elements, representing the broadband characteristics of the
interconnects, are extracted. The response of the proposed SPICE
compatible equivalent-circuit models is shown to be in good agreement
with the frequency-dependent transmission line characteristics of single
and general coupled on-chip interconnects
In this paper, a new finite-difference time-domain (FDTD)
algorithm is proposed in order to eliminate the Courant-Friedrich-Levy
(CFL) condition restraint. The new algorithm is based on an
alternating-direction implicit method. It is shown that the new
algorithm is quite stable both analytically and numerically even when
the CFL condition is not satisfied. Therefore, if the minimum cell size
in the computational domain is required to be much smaller than the
wavelength, this new algorithm is more efficient than conventional FDTD
schemes in terms of computer resources such as central-processing-unit
time. Numerical formulations are presented and simulation results are
compared to those using the conventional FDTD method