Numerical Accuracy of Finite-Difference Methods

Ryohei Ohnishi1, Di Wu1, Takashi Yamaguchi2, Shinichiro Ohnuki1 1College of Science and Technology, Nihon University

1-8-14 Surugadai, Kanda, Chiyoda-ku, Tokyo 101-8308, Japan 2Tokyo Metropolitan Industrial Technology Research Institute

2-4-10, Aomi, Koto-ku, Tokyo 135-0064, Japan

Abstract – The finite-difference methods are widely used for

electromagnetic field analysis in both of time and frequency domains. In this paper, we clarify computational accuracy of the

finite-difference time-domain method and the finite-difference frequency-domain method by analyzing scattering problems of metallic cylinder.

Index Terms — complex frequency domain, finite difference methods, fast inverse Laplace transform method.

1. Introduction

The finite-difference time-domain (FDTD) method [1] and

the finite-difference frequency-domain (FDFD) method [2]

are commonly used for studying electromagnetic wave

responses. In this paper, we investigate scattering problems

of two dimensional metallic cylinder and compare

computational results obtained by the FDTD method and our

recently proposed the finite-difference algorithm.

We have recently developed the finite-difference complex

–frequency-domain (FDCFD) method which extends the

FDFD method to the complex frequency domain in order to

efficiently calculate both of time and frequency responses [3].

2. Computational model and methods

We investigate scattering problems of two dimensional

metallic cylinder in rectangular coordinate system. The

relative permittivity of the cylinder εr is assumed to be the

Lorentz–Drude model [4]. We verify the computational

accuracy of the FDTD and FDFD methods by analyzing the

problems.

In the FDTD method considering the dispersive medium, it

is commonly used the auxiliary differential equation (ADE)

method [5]. In particular, we couple the Maxwell equations

with the following motion equation of electrons to express the

current polarizations:

𝑚(𝑑2𝐮𝑙)

𝑑𝑡2+

𝑚𝜈𝑙(𝑑𝐮𝑙)

𝑑𝑡+ 𝑚𝜔𝑙

2𝐮𝑙 = 𝑞𝐄 (1)

In ADE-FDTD method, electromagnetic fields and current

polarizations are computed by the finite-difference scheme.

Next, our recently developed FDCFD method is briefly

explained [3]. Maxwell’s equations are formulated in

complex frequency (s: =σ+jω) domain, and equations become

the following form:

𝛁 × 𝐇 = 𝑠𝜀𝐄 + 𝐉 (2-a)

𝛁 × 𝐄 = −𝑠𝜇𝐇 (2-b)

For the TE case, the curl equations result in

∂𝐻𝑧

𝜕𝑦= 𝑠𝜀𝐸𝑥 (3-a)

∂𝐻𝑧

𝜕𝑥= −𝑠𝜀𝐸𝑦 (3-b)

∂𝐸𝑦

𝜕𝑦−

∂𝐸𝑥

𝜕𝑦= −𝑠𝜇𝐻𝑧 (3-c)

Using the finite-difference formula, (3) can be rewritten as

[ 𝑠𝜀Δ𝑦 0 −1 1 ⋯ 0

0 𝑠𝜀Δ𝑥−1−1⋱ ⋱ ⋱ ⋮

⋮ ⋱ ⋱ ⋱1 0 𝑠𝜇Δ𝑦 0

0 ⋯ 1 0 𝑠𝜇Δ𝑥]

[ 𝐸𝑥

1

𝐸𝑦1

⋮⋮

𝐻𝑧𝑥𝑛

𝐻𝑧𝑦𝑛 ]

=

[ 𝐽𝑥1

𝐽𝑦1

⋮⋮00 ]

(4)

The size of a unit cell is given by Δx and Δy. In (4), Hzx and

Hzy are components of Hz, which propagate in the x and y

directions, respectively. The unknown electromagnetic field

components in the complex-frequency-domain can be

evaluated using a direct or iterative solver [6], in a process

similar to that of the conventional finite-difference scheme. A

direct solver based on the Thomas algorithm is used to solve

the equations in this paper.

Time-domain responses using FDCFD can be obtained by

collaborating Fast Inverse Laplace Transform (FILT) [7][8] in

Eq. (4). In our algorithm, the exponential function in the

Bromwich integral in (5) is replaced by the cosine hyperbolic

function in (6).

𝑓(𝑡) =1

2𝜋𝑗∫ 𝐹(𝑠)𝑒𝑠𝑡𝑑𝑠

𝛽+∞

𝛽−∞

(5)

𝑒𝑠𝑡 = 𝐸𝑒𝑐 =𝑒𝛼

2 cosh(𝛼 − 𝑠) (6)

By substituting Eq. (5) into the integrand and using the residue

theorem, the approximated time-domain function fec (t, α) can

be evaluated using the following summation:

𝑓𝑒𝑐(𝑡, 𝛼) =𝑒𝛼

𝑡∑(−1)𝑛𝐼𝑚[𝐹(𝑠)]

𝑘

𝑛=1

𝑠 =𝛼 + 𝑗(𝑛 − 0.5)𝜋

𝑡

(7)

where k is the truncation number of an infinite series and F(s)

is the image function of the original time-domain function f(t).

[WeC1-5] 2018 International Symposium on Antennas and Propagation (ISAP 2018)October 23~26, 2018 / Paradise Hotel Busan, Busan, Korea

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Convergence of the alternating series can be accelerated by

adding Euler transformation to Eq. (7) [7]. Here, the accuracy

of fec can be controlled by an approximation parameter α [7].

3. Computational results

The computational model of the metallic cylinder is shown

in Fig.1.The radius of the cylinder is 5nm. The medium is

silver, and the dispersion parameters such as plasma

frequency and damping constant are selected as in [4]. The

incident wave is a plane wave propagating in the x direction

with the wavelength λ = 350nm. We compare the electric field

distribution at 10 fs which is computed by of FDCFD-FILT

and FDTD method. Cell sizes of both methods are Δx = Δy =

0.1nm.

Figs. 2 and 3 show the field distribution of Ey component in

both methods. The result of FDCFD-FILT and that of FDTD

are in excellent agreement.

Finally, we consider accuracy of two finite-difference

methods. Fig.4 shows the relative error between two methods.

The maximum error is about 0.01% occurred in the vicinity of

the surface of the cylinder, and it becomes less than 0.01% in

other places.

4. Conclusions

In this paper, computational accuracy of two finite-

difference methods, FDCFD method and FDTD method, were

verified by studying electric field distribution. The relative

error between FDCFD-FILT and FDTD becames less than

0.01%.

Fig. 1. Computational model of a silver nano-cylinder

Fig. 2. Distribution of Ey component in FDCFD-FILT

Fig. 3. Distribution of Ey component in FDTD

Fig. 4. Relative error of Ey component

Acknowledgements

This work was partly supported by Grant-in-Aid for Scientific

Research (C) (17K06401) and Nihon University College of

Science and Technology Project for Research.

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[8] S. Kishimoto, T. Okada, S. Ohnuki, Y. Ashizawa, and K. Nakagawa,

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10.2528/PIER13081701.

Silver

sylinder r

z

x

y

EH

0 10 20

10

20

x[n

m]

y [nm]

Ey

[V/m

]

-2

2

0

0 10 20

10

20

x[n

m]

y [nm]

Ey

[V/m

]

-2

2

0

0 10 20

10

20

x[n

m]

y [nm]

Relativ

e error [%

]

0

0.010

0.005

2018 International Symposium on Antennas and Propagation (ISAP 2018)October 23~26, 2018 / Paradise Hotel Busan, Busan, Korea

30