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Unconditionally Stable Leapfrog Adi-FDTD Method for Lossy Media
Theng Huat Gan;Eng Leong Tan
PIER M , 2012, DOI: 10.2528/PIERM12090307
Abstract: This paper presents an unconditionally stable threedimensional (3-D) leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method for lossy media. Conductivity terms of lossy media are incorporated into the leapfrog ADI-FDTD method in an analogous manner as the conventional explicit FDTD method since the leapfrog ADI-FDTD method is a perturbation of the conventional explicit FDTD method. Implementation of the leapfrog ADI-FDTD method for lossy media with special consideration for boundary condition is provided. Numerical results and examples are presented to validate the formulation.
Analysis of the Numerical Dispersion of Higher Order ADI-FDTD

Xu Li-jun,Yuan Nai-chang,

电子与信息学报 , 2005,
Abstract: In this paper, a new higher order Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) formulation in particular, a second-order-in-time, fourth-order-in-space AD-FDTD method is presented for the first time. At the same time ,the unconditional stability of the higher order ADI-FDTD formulation is analytically proved. By analysis of the amplification factors, the numerical dispersion relation is derived. In addition, the numerical dispersion errors are investigated. Finally numerical results indicate that the higher order ADI-FDTD has s better accuracy compared to the ADI-FDTD method.
PML Implementation for ADI-FDTD in Dispersive Media

Wang Yu,Yuan Nai-chang,
王 禹

电子与信息学报 , 2005,
Abstract: Alternating Direction Implicit-Finite Difference Time Domain(ADI-FDTD) is unconditionally stable and the maximum time step size is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step size of ADI-FDTD can be enlarged arbitrarily. In this paper 2D PML implementation is proposed for ADI-FDTD in dispersive media using recursive convolution method. ADI-FDTD formulations for dispersive media can be derived from the simplified Perfectly Matched Layer (PML). Numerical results of ADI-FDTD with PML for dispersive media are compared with FDTD. Good agreement is observed.
The Stability and Numerical Dispersion Study of the ADI-SFDTD Algorithm  [cached]
Zongxin Liu,Yiwang Chen,Xin Xu,Xuegang Sun
Modern Applied Science , 2011, DOI: 10.5539/mas.v5n6p157
Abstract: In this letter, the alternating-direction-implicit (ADI) technique is applied to Symplectic finite-difference time-domain (SFDTD) method, the curl operator is endued with two different styles when doing computation from the th progression to th progression. It holds the advantages of both ADI-FDTD and SFDTD, not only eliminating the restriction of the Courant-Friedrich-Levy (CFL), but also holding the inner characteristics of Maxwell’s equations. The analytical accuracy and efficiency of the proposed method is verified good.

Zhou Xiaojun,Yu Zhiyuan,Lin Weigan,

电子与信息学报 , 2000,
Abstract: This paper proposes a new algorithm based on conformal mapping and FDTD method, and derives the numerical stability and numerical dispersion equations of conformal mapping FDTD algorithm. As an example, the relative errors of numerical wavelengths for TE modes in a circular waveguide in different cell number are calculated. The errors for different propagation constants and for different radius semicircle electric wall approaches to singularity at origin are analyzed. By selecting cell number appropriately, high accuracy can be obtained.
Stability and conservation properties of transient field simulations using FIT  [PDF]
R. Schuhmann,T. Weiland
Advances in Radio Science : Kleinheubacher Berichte , 2003,
Abstract: Time domain simulations for high-frequency applications are widely dominated by the leapfrog timeintegration scheme. Especially in combination with the spatial discretization approach of the Finite Integration Technique (FIT) it leads to a highly efficient explicit simulation method, which in the special case of Cartesian grids can be regarded to be computationally equivalent to the Finite Difference Time Domain (FDTD) algorithm. For stability reasons, however, the leapfrog method is restricted to a maximum stable time step by the well-known Courantcriterion, and can not be applied to most low-frequency applications. Recently, some alternative, unconditionally stable techniques have been proposed to overcome this limitation, including the Alternating Direction Implicit (ADI)-method. We analyze such schemes using a transient modal decomposition of the electric fields. It is shown that stability alone is not sufficient to guarantee correct results, but additionally important conservation properties have to be met. Das Leapfrog-Verfahren ist ein weit verbreitetes Zeitintegrationsverfahren für transiente hochfrequente elektrodynamischer Felder. Kombiniert mit dem r umlichen Diskretisierungsansatz der Methode der Finiten Integration (FIT) führt es zu einer sehr effizienten, expliziten Simulationsmethode, die im speziellen Fall kartesischer Rechengitter als quivalent zur Finite Difference Time Domain (FDTD) Methode anzusehen ist. Aus Stabilit tsgründen ist dabei die Zeitschrittweite durch das bekannte Courant-Kriterium begrenzt, so dass das Leapfrog- Verfahren für niederfrequente Probleme nicht sinnvoll angewendet werden kann. In den letzten Jahren wurden alternativ einige andere explizite oder “halb-implizite" Zeitbereichsverfahren vorgeschlagen, u.a. das “Alternating Direction Implicit" (ADI)-Verfahren, die keiner Beschr nkung des Zeitschritts aus Stabilit tsgründen unterliegen. Es zeigt sich aber, dass auch diese Methoden im niederfrequenten Fall nicht zu sinnvollen Simulationsergebnissen führen. Wie anhand einer transienten Modalanalyse der elektrischen Felder in einem einfachen 2D-Beispiel deutlich wird, ist die Ursache dafür die Verletzung wichtiger physikalischer Erhaltungseigenschaften durch ADI und verwandte Methoden.
Energy Identities of ADI-FDTD Method with Periodic Structure  [PDF]
Rengang Shi, Haitian Yang
Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.62025
Abstract: In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L2 and H1 norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.
Application of Artificial Anisotropy in 3-D ADI-FDTD Method

Zhang Yan,,Shan-wei,
张 岩

电子与信息学报 , 2006,
Abstract: Attention is focused on a new method to reduce the numerical dispersion of the 3-D Alternating-Direction Implicit Finite-Difference Time-Domain(ADI-FDTD) method through artificial anisotropy. As the wave propagation can be speeded up by introducing proper anisotropy parameters into the 3-D ADI-FDTD method, the numerical dispersion can be reduced and the accuracy can be improved significantly. First, the numerical formulations of the 3-D ADI-FDTD method are modified. Secondly, the new numerical dispersion relation is derived. And consequently the relative permittivity tensor of artificial anisotropy can be obtained. In order to demonstrate the accuracy and efficiency of this new method, a hollow waveguide and a waveguide with discontinuous structure are simulated as examples. In addition the reduction of numerical dispersion is investigated as a function of the relative permittivity tensor of artificial anisotropy. Furthermore, the numerical results and the computational requirements of the proposed method are compared with those of the conventional 3-D ADI-FDTD method. It is found that this new method is accurate and efficient.
A Novel 3-D Weakly Conditionally Stable FDTD Algorithm
Jian-Bao Wang;Bi-Hua Zhou;Li-Hua Shi;Cheng Gao;Bin Chen
PIER , 2012, DOI: 10.2528/PIER12071904
Abstract: For analyzing the electromagnetic problems with the fine structures in one or two directions, a novel weakly conditionally stable finite-difference time-domain (WCS-FDTD) algorithm is proposed. By dividing the 3-D Maxwell's equations into two parts, and applying the Crank-Nicolson (CN) scheme to each part, a four sub-step implicit procedures can be obtained. Then by adjusting the operational order of four sub-steps, a novel 3-D WCS-FDTD algorithm is derived. The proposed method only needs to solve four implicit equations, and the Courant-Friedrich-Levy (CFL) stability condition of the proposed algorithm is more relaxed and only determined by one space discretisation. In addition, numerical dispersion analysis demonstrates the numerical phase velocity error of the weakly conditionally stable scheme is less than that of the 3-D ADI-FDTD scheme.
An Efficient Method to Reduce the Numerical Dispersion in the HIE-FDTD Scheme  [PDF]
Juan Chen, Anxue Zhang
Wireless Engineering and Technology (WET) , 2011, DOI: 10.4236/wet.2011.21005
Abstract: A parameter optimized approach for reducing the numerical dispersion of the 3-D hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) is presented in this letter. By adding a parameter into the HIE-FDTD formulas, the error of the numerical phase velocity can be controlled, causing the numerical dispersion to decrease significantly. The numerical stability and dispersion relation are presented analytically, and numerical experiments are given to substantiate the proposed method.
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