LD comparison of LD FDTD method and AH FDTD method.
Abstract
The orthogonal expansion in timedomain method is a new kind of unconditionally stable finitedifference timedomain (FDTD) method for solving the Maxwell equation efficiently. Generally, it can be implemented by two schemes: marchingoninorder and parallelinginorder, which, respectively, use weighted Laguerre polynomials and associated Hermite functions as temporal expansions and testing functions. This chapter summarized parallelinginorderbased FDTD method using associated Hermite functions and Legendre polynomials. And a comparison from theoretical analysis to numerical examples is shown. The LD integral transfer matrix can be considered as a “dual” transformation for AH differential matrix, which gives a possible way to find more potential orthogonal basis function to implement a parallelinginorder scheme. In addition, the differences with these two orthogonal functions are also analyzed. From the numerical results, we can see their agreements in some general cases while differing in some cases such as shielding analysis with the longtime response requirement.
Keywords
 associated Hermite
 finitedifference timedomain (FDTD)
 Legendre polynomials
 parallelinginorder
 unconditionally stable
1. Introduction
To overcome the numerical stability constraints of conventional finitedifference timedomain (FDTD) method [1, 2], many unconditionally stable methods to reduce or eliminate requirements of the stability condition have been proposed and developed, such as alternatingdirection implicit method [2, 3] and locally onedimensional schemes [3], explicit and unconditionally stable FDTD method [4], and orthogonal expansions in time domain [5, 6, 7, 8]. For the orthogonal expansions schemes, fieldversustime variations in the FDTD space lattice are expanded using an appropriate set of orthogonal temporal basis and testing functions, such as weighted Laguerre polynomials (WLP) and associated Hermite (AH) functions, which leads to two different solution schemes: marchingoninorder and parallelinginorder, respectively. Both of them appear to be promising according to the reported work where the computational time can be reduced to at least 10% of the conventional FDTD scheme [1]. Recently, the Legendre (LD) polynomials are explored as another possible orthogonal expansion incorporated with FDTD to form a parallelinginorderbased unconditionally stable FDTD method. Based on it, in this chapter, we made a comparison investigation for these two new methods, which are AH FDTD method and LD FDTD method, especially focused on their differences. Through a numerical example, we validate their effectiveness when compared with the conventional FDTD method and summarized the characteristics of the two methods.
2. Formulation for parallelinginorder scheme: AH and LD functions
2.1 2D Maxwell’s equations in time domain
The 2D timedomain Maxwell’s equations with the TEz wave case in lossy medium are considered:
where
2.2 The differential and integral transfer matrices to deal with the partial differential term in Maxwell’s equations
2.2.1 The associated Hermite function
Associated Hermite function is defined as
where
From [7], if a causal function
we can deduce the first derivative of
Then, the Qtuple AH domain coefficients for
where
By using (8), the partial differential term in Maxwell?s equations can readily be dealt with, and finally, a fivepoint banded matrix equation for Hz component can be obtained [9].
2.2.2 The associated Legendre polynomial
We expand all the temporal quantities in terms of the associated Legendre polynomial given by [10]:
where
and
where
From the intrinsic features of Legendre function, the differential relationship can be described as
If the field derivative of
where
Connecting (15) and (11), we can get
When assembling
where
Alternatively, Eq. (17) can be rewritten as.
2.3 From time domain to orthogonal domain and reconstruction
When the differential or integral transfer matrices are obtained, the timedomain Maxwell equation can be transformed directly into AH or LD domain. Here, let us set LD as an example to illustrate the later formulation.
Similar to the parallelinginorderbased AH FDTD method, we can apply a Qtupledomain transformation for LD FDTD method to (1)–(3) and discretize them as the following:
where
where
where
By using eigenvalue transformation from
where
3. Comparison for the two methods
The above formula can be regarded and classified as a uniform OF differential transfer matrix transformation. Therefore, as long as the LD differential matrix is replaced by the AH domain differential transfer matrix, the FDTD algorithm based on the LD orthogonal basis function, LD FDTD, including the parallel solution AH FDTD algorithm [9], and the alternate direction efficient calculation [11] can be easily realized. The implementation of the program only requires a simple modification.
Table 1 gives a comparison of the relevant properties of the LD FDTD method and the AH FDTD method. It can be seen that the two methods can be considered as a “dual” system, because the AH differential matrix is the basic element of the AH FDTD method and the LD integration matrix is also the basic element of the LD FDTD method. This gives us a revelation that is it possible that any orthogonal basis function can construct a differential or integral transfer matrix and then easily implement a parallelinginorder scheme similar like AH FDTD algorithm? The answer might be
AH FDTD  LD FDTD 

Differential transfer matrix  Integral transfer matrix 







Scale factor l = TQ Finite order of Q 
With timefrequency Homomorphism  Without timefrequency Homomorphism 
Antisymmetry Eigenvalue conjugate symmetry 
Antisymmetry Eigenvalue conjugate symmetry 


4. Numerical verification
4.1 An infinitely large lossy dielectric plate
As AH or LD FDTD method shares with almost the same program, a 1D program is set for a general verification. Figure 1 shows the simulation results when a uniform plane wave penetrates an infinitely large lossy dielectric plate. The figure includes the electric field waveforms calculated by the AH FDTD method and the LD FDTD method and their relative errors with respect to the conventional FDTD method. It can be seen that the timedomain waveforms of both can be consistent with the results of the FDTD method and the relative errors are basically the same, only differing in the initial part. Therefore, in general, when the order of the two basic functions is the same and the parameters are selected reasonably, the accuracy is basically the same, and the efficiency is almost the same.
4.2 An nonuniform parallel plate waveguide with a slot
However, the two methods also have the differences when simulating the longtime response applications, such as the example in [12]. The numerical example is set as a TEz wave propagation in a parallel plate waveguide, as shown in Figure 2. It is with a PEC slot of the thickness 0.2 mm and the distance 0.2 mm and a partly filled dielectric material of the thickness 0.8 mm with the dielectric medium parameters given as two cases: case I, ε = 11 ε_{0}, μ = μ_{0}, σ_{e} = 0.003 S/m, and σ_{m} = 0 Ω/m; case II, ε = 2 ε_{0}, μ = μ_{0}, σ_{e} = 30,000 S/m, and σ_{m} = 0 Ω/m. There are 140 × 8 uniform cells (
where
The Ey electric field responses at measurement point p1 and p2, located at the center of the slot and behind the medium, respectively, are calculated, which are both in agreement with the conventional FDTD method as shown in Figures 2 and 3. For comparison, the AH FDTD method is also used in these two cases. One can find the good results in Figure 3, but the errors come out in Figure 4 for AH FDTD method when the same number of orthogonal functions (Q = 80 for case I or 300 for case II) is used as LD FDTD method. However, when Q reaches 800, the results from AH FDTD method can achieve a comparable accuracy with the ones from LD FDTD method. One should note that for case II the waveform at point p2 has larger amplitude attenuation and longer delay than the result at point p1 due to the high dielectric medium located between them.
Tables 2 and 3 show the comparison of the computational resources. We can see that the simulation takes much more time for the FDTD method compared with proposed method, especially for the case of II, while the tradeoff for the proposed method is that it consumes more memory than conventional FDTD method, which is similar to the AH FDTD method. In addition, from Table 3, we can find the advantages compared with AH FDTD method that the proposed method can use relative smaller memory storage and slightly fewer CPU times to get a readily results.
Δt (ps)  Memory (MB)  CPU time (s)  

FDTD (N = 6000)  0.21  1.8  2.97 
AH FDTD (Q = 80)  21  2.9  1.32 
LD FDTD (Q = 80)  21  2.9  1.32 
Δt (ps)  Memory (MB)  CPU time (s)  

FDTD (N = 60,000)  0.21  1.8  30.8 
AH FDTD (Q = 300)  21  11.8  1.55 
AH FDTD (Q = 800)  21  28.9  1.95 
LD FDTD (Q = 300)  21  11.8  1.55 
5. Conclusions and future developments
The parallelinginorderbased unconditionally stable FDTD methods are introduced using associated Hermite and Legendre polynomials in this chapter. The direct Qtupledomain transformation for timedomain Maxwell equation is guaranteed by using the integral matrix and differential matrix for Legendre function and associated Hermite functions that are introduced from the intrinsic integral or differential features for these orthogonal functions. Normally, the integral matrix of Legendre function can be considered as an inverse relationship from the differential operator, similar to the AH differential matrix. From this view, we can consider them as a uniform algorithm organized from the parallelinginorder solution scheme. In addition, this chapter also detailed the different properties and the formula with these two methods theoretically and tested by numerical examples. Numerical examples for 1D and 2D cases validate their effectiveness and show LD FDTD with a better performance than AH FDTD method, in longtime simulation applications. In the next step, the more general parallelinginorder scheme should be summarized, and then find or construct other possible orthogonal functions for their specific applications.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants 61801217 and 51477183 and Natural Science Foundation of Jiangsu Province under Grant BK20180422. This support is gratefully acknowledged.
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