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An unconditionally stable scheme for the finite-difference time-domain method

An unconditionally stable scheme for the finite-difference time-domain method,10.1109/TMTT.2003.808732,IEEE Transactions on Microwave Theory and Techn

An unconditionally stable scheme for the finite-difference time-domain method   (Citations: 59)
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In this work, we propose a numerical method to obtain an unconditionally stable solution for the finite-difference time-domain (FDTD) method for the TEz case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the time-domain 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 marching-on 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 domain-weighted Laguerre polynomials for the expansion of the temporal variation of the fields, the spatial and the temporal variables can be separated.
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    • ...Recently, a new unconditionally stable FDTD method with weighted Laguerre polynomials (WLP-FDTD) [7]‐[9] was introduced...
    • ...It can eliminate the restriction of the CFL condition, without the special treatment for the additional terms, and only a limited number of iterations are needed to achieve a satisfactory accuracy [7]...
    • ...A set of orthogonal basis functions can be constructed using Laguerre Polynomials [7], given by...
    • ...According to the derivational procedure in [7], (1)‐(3) can be written as...
    • ...Meanwhile, only a limited number of iterations are needed to achieve a satisfactory accuracy [7]...
    • ...The upper limit of infinity can be replaced by a finite time interval [7]...

    Zhao-Yang Caiet al. The WLP-FDTD Method for Periodic Structures With Oblique Incident Wave

    • ...On the other hand, another unconditionally stable scheme were intelligently developed [5], [6] for the conventional gridbased finite-difference time-domain (FDTD) method; there the weighted decaying Laguerre polynomials are used as temporal basis and testing functions to expand unknown field quantities and to minimize residual errors in the time domain; then a procedure of the Method of Moments or Method of Weighted Residuals as described ...
    • ...The method has been shown to be efficient due to the fact that its solutions can be computed in a recursive march-in-order manner [5]...
    • ...Therefore, an orthogonal set of basis functions can be formed as [5]:...
    • ...Scaling factor is used to increase or decrease the support provided by the above expansion as described in [5]...
    • ...For any expanded field component function (with being , and ), its first-order temporal derivative with respect to time is [5]:...
    • ...Upper limit is chosen in such a way that the waveform of source has practically decayed to zero after [5]...
    • ...The value of can be determined by [5], [8], where is the temporal duration to be simulated and is the bandwidth of the signal of interest...
    • ...Then it can be stored for the subsequent computations as described in [5]...
    • ...In both examples, the modulated Gaussian pulse below is chosen as an exciting electric current source, the same as the one used in [5]:...

    Xiaojie Chenet al. An Unconditionally Stable Radial Point Interpolation Meshless Method W...

    • ...The time span is chosen in such a way that the waveformsofinteresthavepracticallydecayedtozero[8].Forconvenience,wedefine asetof auxiliary matricesas(19)‐(24), shown at the bottom of the following page...

    Yan-Tao Duanet al. Efficient Implementation for 3-D Laguerre-Based Finite-Difference Time...

    • ...In order to eliminate the stability condition constraint, a new unconditionally stable FDTD method with weighted Laguerre polynomials (WLPs) can be used [2]...
    • ...The weighted Laguerre functions are given as [2]...
    • ...where . Insert (6) and (7) into (1)–(4), According to the derivational procedure in [2], (1)–(4) can be written as...
    • ...The update equations of the magnetic field components can be derived from (10) and (11) according to the procedure described in [2]...
    • ...In an actual simulation, the expansion coefficients of the electric field components must be updated as following sequence: , and . Then, one can reconstruct the field components in the time domain according to the method in [2]...
    • ...The agreement between FDTD, conventional WLP-FDTD [2], and efficient WLP-FDTD is very good...

    Yan-Tao Duanet al. PML Absorbing Boundary Condition for Efficient 2-D WLP-FDTD Method

    • ...To over come this difficulty, an unconditionally stable FDTD with weighted Laguerre polynomials (WLP–FDTD) was proposed [4]...
    • ...With reference to Chung et al.[4], using the basis weighted...

    Yin Qinet al. An unconditionally stable FDTD with weighted Laguerre polynomials in c...

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