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44
A split step approach for the 3D Maxwell’s equations
 J. Comput. Appl. Math
, 2003
"... Abstract. Splitstep procedures have previously been used successfully in a number of situations, e.g. for Hamiltonian systems, such as certain nonlinear wave equations. In this study, we note that one particular way to write the 3D Maxwell’s equations separates these into two parts, requiring in a ..."
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Abstract. Splitstep procedures have previously been used successfully in a number of situations, e.g. for Hamiltonian systems, such as certain nonlinear wave equations. In this study, we note that one particular way to write the 3D Maxwell’s equations separates these into two parts, requiring in all only the solution of six uncoupled 1D wave equations. The approach allows arbitrary orders of accuracy in both time and space, and features in many cases unconditional stability. 1.
An unconditionally stable scheme for the finitedifference timedomain method
 IEEE Trans. Microwave Theory Tech
, 2003
"... Abstract—In this work, we propose a numerical method to obtain an unconditionally stable solution for the finitedifference timedomain (FDTD) method for the „i case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the ti ..."
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Abstract—In this work, we propose a numerical method to obtain an unconditionally stable solution for the finitedifference timedomain (FDTD) method for the „i case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the timedomain Maxwell’s equations by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives can be handled analytically, which results in an implicit relation. In this way, the time variable is eliminated from the computations. By introducing the Galerkin temporal testing procedure, the marchingon in time method is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials if the input waveform is of arbitrary shape. Since the weighted Laguerre polynomials converge to zero as time progresses, the electric and magnetic fields when expanded in a series of weighted Laguerre polynomials also converge to zero. The other novelty of this approach is that, through the use of the entire domainweighted Laguerre polynomials for the expansion of the temporal variation of the fields, the spatial and the temporal variables can be separated. To verify the accuracy and the efficiency of the proposed method, we compare the results of the conventional FDTD method with the proposed method. Index Terms—Finite difference time domain (FDTD), Laguerre polynomials, unconditionally stable scheme. I.
Efficient numerical methods for simulation of highfrequency active devices
 IEEE Trans. Microw. Theory Tech
, 2006
"... Abstract—We present two new numerical approaches for physical modeling of highfrequency semiconductor devices using filterbank transforms and the alternatingdirection implicit finitedifference timedomain method. In the first proposed approach, a preconditioner based on the filterbank and wavele ..."
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Abstract—We present two new numerical approaches for physical modeling of highfrequency semiconductor devices using filterbank transforms and the alternatingdirection implicit finitedifference timedomain method. In the first proposed approach, a preconditioner based on the filterbank and wavelet transforms is used to facilitate the iterative solution of Poisson’s equation and the other semiconductor equations discretized using implicit schemes. The second approach solves Maxwell’s equations which, in conjunction with the semiconductor equations, describe the complete behavior of highfrequency active devices, with larger timestep size. These approaches lead to the significant reduction of the fullwave simulation time. For the first time, we can reach over 95 % reduction in the simulation time by using these two techniques while maintaining the same degree of accuracy achieved using the conventional approach. Index Terms—Alternatingdirection implicit finitedifference timedomain (ADIFDTD) method, filterbank transforms, fullwave analysis, global modeling, highfrequency devices, preconditioning. I.
Investigation of Numerical TimeIntegrations of the Maxwell's Equations Using the Staggered Grid Spatial Discretization
, 2002
"... The Yeemethod is a simple and elegant way of solving the timedependent Maxwell's equations. On the other hand this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible timesteps. This is why, during the last decade, th ..."
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The Yeemethod is a simple and elegant way of solving the timedependent Maxwell's equations. On the other hand this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible timesteps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the timestep can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yeemethod. Three other numerical methods are discussed: the NamikiZhengChenZhang (NZCZ) ADI method, the KoleFiggede Raedtmethod (KFR) and a Krylovspace method. All methods are discussed with nonhomogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZmethod without any computer algebraic tool. Moreover, we accelerate the Krylovspace method in the approximation of the matrix exponential with a skewsymmetric formulation of the semidiscretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in 1D numerical tests. Index Terms FDTD Method, Stability, Unconditional Stability I.
MODELING THE INTERACTION OF TERAHERTZ PULSE WITH HEALTHY SKIN AND BASAL CELL CAR CINOMA USING THE UNCONDITIONALLY STABLE FUNDAMENTAL ADIFDTD METHOD
"... Abstract—This paper presents the application of unconditionally stable fundamental finitedifference timedomain (FADIFDTD) 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 ..."
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Abstract—This paper presents the application of unconditionally stable fundamental finitedifference timedomain (FADIFDTD) 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 FADIFDTD method. Numerical experiments on delineating the BCC margin from healthy skin are demonstrated using the FADIFDTD method based on reflected terahertz pulse. Hence, the FADIFDTD 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. 1.
Some numerical techniques for Maxwell’s equations in different types of geometries
 in Topics in Computational Wave Propagation
, 2003
"... Abstract. Almost all the difficulties that arise in finite difference time domain solutions of Maxwell’s equations are due to material interfaces (to which we include objects, such as antennas, wires, etc.) Different types of difficulties arise if the geometrical features are much larger than or muc ..."
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Abstract. Almost all the difficulties that arise in finite difference time domain solutions of Maxwell’s equations are due to material interfaces (to which we include objects, such as antennas, wires, etc.) Different types of difficulties arise if the geometrical features are much larger than or much smaller than a typical wave length. In the former case, the main difficulty has to do with the spatial discretization, which needs to combine good geometrical flexibility with a relatively high order of accuracy. After discussing some options for this situation, we focus on the latter case. The main problem here is to find a time stepping method which combines a very low cost per time step with unconditional stability. The first such method was introduced in 1999 and is based on the ADI principle. We will here discuss that method and some subsequent developments in this area. Key words. Maxwell’s equations, FDTD, ADI, split step, pseudospectral methods, finite differences, spectral elements.
Tovbis Linear Algebra and Applications
, 1992
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are
An Implicit Characteristic Based Method for Electromagnetics
, 2001
"... An implicit characteristicbased approach for numerical solution of Maxwell's timedependent curl equations in flux conservative form is introduced. This method combines a characteristic based finite difference spatial approximation with an implicit lowerupper approximate factorization (LU/AF) ..."
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An implicit characteristicbased approach for numerical solution of Maxwell's timedependent curl equations in flux conservative form is introduced. This method combines a characteristic based finite difference spatial approximation with an implicit lowerupper approximate factorization (LU/AF) time integration scheme. This approach is advantageous for threedimensional applications because the characteristic differencing enables a twofactor approximate factorization that retains its unconditional stability in three space dimensions, and it does not require solution of tridiagonal systems. Results are given both for a Fourier analysis of stability, damping and dispersion properties, and for onedimensional model problems involving propagation and scattering for free space and dielectric materials using both uniform and nonuniform grids. The explicit FDTD algorithm is used as a convenient reference algorithm for comparison. The onedimensional results indicate that for low frequency problems on a highly resolved uniform or nonuniform grid, this LU/AF algorithm can produce accurate solutions at Courant numbers significantly greater than one, with a corresponding improvement in efficiency for simulating a given period of time. This approach appears promising for development of LU/AF schemes for three dimensional applications.
A wideband ADIFDTD algorithm for the design of double negative metamaterialbased waveguides and antenna substrates
 IEEE Trans. Magn
, 2007
"... An enhanced 3D alternatingdirection implicit finitedifference timedomain method for the systematic analysis of double negative metamaterialbased waveguide and antenna devices is presented in this paper. Being fully frequencydependent, the new scheme introduces a set of generalized multidirect ..."
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An enhanced 3D alternatingdirection implicit finitedifference timedomain method for the systematic analysis of double negative metamaterialbased waveguide and antenna devices is presented in this paper. Being fully frequencydependent, the new scheme introduces a set of generalized multidirectional operators which incorporate the suitable Lorentz–Drude model and subdue inherent lattice deficiencies for broad spectra. So dispersion errors, as timesteps exceed the Courant limit, are drastically minimized yielding fast and accurate solutions for propagating and evanescent waves. The technique is applied in the design of microwave structures, realized via the prior model or networks of thin wires and splitring resonators. Numerical results certify its merits, without requiring elongated simulations and excessive overheads. Index Terms—ADIFDTD method, antennas, double negative (DNG) media, metamaterials, splitring resonators, waveguides. I.
STUDY ON THE STABILITY AND NUMERICAL ERROR OF THE FOURSTAGES SPLITSTEP FDTD METHOD INCLUDING LUMPED INDUCTORS
"... Abstract—The stability and numerical error of the extended fourstages splitstep finitedifference timedomain (SS4FDTD) method including lumped inductors are systematically studied. In particular, three different formulations for the lumped inductor are analyzed: the explicit, the semiimplicit, a ..."
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Abstract—The stability and numerical error of the extended fourstages splitstep finitedifference timedomain (SS4FDTD) method including lumped inductors are systematically studied. In particular, three different formulations for the lumped inductor are analyzed: the explicit, the semiimplicit, and the implicit schemes. Then, the numerical stability of the extended SS4FDTD method is analyzed by using the von Neumann method, and the results show that the proposed method is unconditionallystable in the semiimplicit 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 SS4FDTD 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 semiimplicit and implicit schemes. Finally, a simple microstrip circuit including a lumped inductor is simulated to demonstrate the validity of the theoretical results. 1.