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Two Efficient Unconditionally-Stable Four-Stages Split-Step FDTD Methods with Low Numerical Dispersion
Yong-Dan Kong;Qing-Xin Chu;Rong-Lin Li
PIER B , 2013, DOI: 10.2528/PIERB12103011
Abstract: Two efficient unconditionally-stable four-stages split-step (SS) finite-difference time-domain (FDTD) methods based on controlling parameters are presented, which provide low numerical dispersion. Firstly, in the first proposed method, the Maxwell's matrix is split into four sub-matrices. Simultaneously, two controlling parameters are introduced to decrease the numerical dispersion error. Accordingly, the time step is divided into four sub-steps. The second proposed method is obtained by adjusting the sequence of the sub-matrices deduced in the first method. Secondly, the theoretical proofs of the unconditional stability and dispersion relations of the proposed methods are given. Furthermore, the processes of obtaining the controlling parameters for the proposed methods are shown. Thirdly, the dispersion characteristics of the proposed methods are also investigated, and numerical dispersion errors of the proposed methods can be decreased significantly. Finally, to substantiate the efficiency of the proposed methods, numerical experiments are presented.
Study on the Stability and Numerical Error of the Four-Stages Split-Step FDTD Method Including Lumped Inductors
Yong-Dan Kong;Qing-Xin Chu;Rong-Lin Li
PIER B , 2012, DOI: 10.2528/PIERB12062008
Abstract: The stability and numerical error of the extended four-stages split-step finite-difference time-domain (SS4-FDTD) method including lumped inductors are systematically studied. In particular, three different formulations for the lumped inductor are analyzed: the explicit, the semi-implicit, and the implicit schemes. Then, the numerical stability of the extended SS4-FDTD method is analyzed by using the von Neumann method, and the results show that the proposed method is unconditionally-stable in the semi-implicit and the implicit schemes, whereas it is conditionally stable in the explicit scheme, which its stability is related to both the mesh size and the values of the element. Moreover, the analysis of the numerical error of the extended SS4-FDTD is studied, which is based on the Norton equivalent circuit. Theoretical results show that: 1) the numerical impedance is a pure imaginary for the explicit scheme; 2) the numerical equivalent circuit of the lumped inductor is an inductor in parallel with a resistor for the semi-implicit and implicit schemes. Finally, a simple microstrip circuit including a lumped inductor is simulated to demonstrate the validity of the theoretical results.
High-Order Unconditionally-Stable Four-Step Adi-FDTD Methods and Numerical Analysis
Yong-Dan Kong;Qing-Xin Chu;Rong-Lin Li
PIER , 2013, DOI: 10.2528/PIER12102205
Abstract: High-order unconditionally-stable three-dimensional (3-D) four-step alternating direction implicit finite-difference time-domain (ADI-FDTD) methods are presented. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four sub-procedures. Accordingly, the time step is divided into four sub-steps. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method are used to approximate the spatial differential operators first, and then the uniform formulation of the proposed high-order schemes is generalized. Subsequently, the analysis shows that all the proposed high-order methods are unconditionally stable. The generalized form of the dispersion relations of the proposed high-order methods is carried out. Finally, in order to demonstrate the validity of the proposed methods, numerical experiments are presented. Furthermore, the effects of the order of schemes, the propagation angle, the time step, and the mesh size on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed schemes are lower than that of the traditional ADI-FDTD method.
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.
Weakly Conditionally Stable and Unconditionally Stable FDTD Schemes for 3D Maxwell's Equations
Juan Chen;Jianguo Wang
PIER B , 2010, DOI: 10.2528/PIERB09110502
Abstract: To overcome the Courant limit on the time step size of the conventional finite-difference time-domain (FDTD) method, some weakly conditionally stable and unconditionally stable FDTD methods have been developed recently. To analyze the relations between these methods theoretically, they are all viewed as approximations of the conventional FDTD scheme in present discussion. The errors between these methods and the conventional FDTD method are presented analytically, and the numerical performances, including computation accuracy, efficiency, and memory requirements, are discussed, by comparing with those of the conventional FDTD method.
Gpu Accelerated Unconditionally Stable Crank-Nicolson FDTD Method for the Analysis of Three-Dimensional Microwave Circuits
Kan Xu;Zhenhong Fan;Da-Zhi Ding;Ru-Shan Chen
PIER , 2010, DOI: 10.2528/PIER10020606
Abstract: The programmable graphics processing unit (GPU) is employed to accelerate the unconditionally stable Crank-Nicolson finite-difference time-domain (CN-FDTD) method for the analysis of microwave circuits. In order to efficiently solve the linear system from the CN-FDTD method at each time step, both the sparse matrix vector product (SMVP) and the arithmetic operations on vectors in the bi-conjugate gradient stabilized (Bi-CGSTAB) algorithm are performed with multiple processors of the GPU. Therefore, the GPU based BI-CGSTAB algorithm can significantly speed up the CN-FDTD simulation due to parallel computing capability of modern GPUs. Numerical results demonstrate that this method is very effective and a speedup factor of 10 can be achieved.
Controlling the accuracy of unconditionally stable algorithms in Cahn-Hilliard Equation  [PDF]
Mowei Cheng,James A. Warren
Physics , 2006, DOI: 10.1103/PhysRevE.75.017702
Abstract: Given an unconditionally stable algorithm for solving the Cahn-Hilliard equation, we present a general calculation for an analytic time step $\d \tau$ in terms of an algorithmic time step $\dt$. By studying the accumulative multi-step error in Fourier space and controlling the error with arbitrary accuracy, we determine an improved driving scheme $\dt=At^{2/3}$ and confirm the numerical results observed in a previous study \cite{Cheng1}.
An Unconditional Stable 1D-FDTD Method for Modeling Transmission Lines Based on Precise Split-Step Scheme
Wei Wang;Pei-Guo Liu;Yu-Jian Qin
PIER , 2013, DOI: 10.2528/PIER12103007
Abstract: his paper presented a novel unconditional stable FDTD (US-FDTD) algorithm for solving the transient response of uniform or nonuniform multiconductor transmission line with arbitrary coupling status. Analytical proof of unconditional stability and detailed analysis of numerical dispersion are presented. The precise split-time-step scheme has been introduced to eliminate the restriction of the Courant-Friedrich-Levy (CFL) condition. Compared to the conventional USFDTD methods, the proposed approach generally achieves lower phase velocity error for coarse temporal resolution. So larger time scales can be chosen for the transient simulation to achieve accurate results efficiently. Several examples of coupled uniform and nonuniform lines are presented to demonstrate the accuracy, stability, and efficiency of the proposed model.
Modeling the Interaction of Terahertz Pulse with Healthy Skin and Basal Cell Carcinoma Using the Unconditionally Stable Fundamental Adi-FDTD Method
Ding Yu Heh;Eng Leong Tan
PIER B , 2012, DOI: 10.2528/PIERB11090905
Abstract: This paper presents the application of unconditionally stable fundamental finite-difference time-domain (FADI-FDTD) method in modeling the interaction of terahertz pulse with healthy skin and basal cell carcinoma (BCC). The healthy skin and BCC are modeled as Debye dispersive media and the model is incorporated into the FADI-FDTD method. Numerical experiments on delineating the BCC margin from healthy skin are demonstrated using the FADI-FDTD method based on reflected terahertz pulse. Hence, the FADI-FDTD method provides further insight on the different response shown by healthy skin and BCC under terahertz pulse radiation. Such understanding of the interaction of terahert pulse radiation with biological tissue such as human skin is an important step towards the advancement of future terahertz technology on biomedical applications.
Transient Analysis of Transmission Line with Arbitrary Loads Based on the Split-step Crank-Nicolson-FDTD Method

Wang Wei Zhou Dong-ming Liu Pei-guo Qin Yu-jian,

电子与信息学报 , 2013,
Abstract: A novel Crank-Nicolson (CN)-FDTD method based on the split-step scheme is proposed in this paper, so as to calculate the electromagnetic transients in transmission line with complex circuit terminals accurately and efficiently. An analytical proof of unconditional stability of the method is provided. Combined with the hybrid one-port equivalent model, the transmission system is decomposed into lumped and distributed portions independently. It can solve the time response of the complex circuit networks by utilizing the Modified Nodal Analysis (MNA) method. Unlike the former methods, the maximum time step size is not limited by the restriction of Courant-Friedrichs-Lewy (CFL) stability constraint. In addition, the dispersion errors can be reduced by the precision sub-time-step scheme. The method is utilized to the transient analysis of the single transmission line. The results show that the proposed method provides higher efficiency and good stability under the same precision level.
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