3. Integral form of Maxwell's equation and a correction to the FDTD method

The FDTD method is a numerical method for solving Maxwell's equations using a computer. Any computer can have a finite number of degrees of freedom because it has a finite memory size. In contrast, an electromagnetic field in continuum space has infinitely many degrees of freedom, because the field exists at every point of the space–time continuum. Therefore, Maxwell's equations must be suitably approximated for us to be able to calculate them using a computer. The algorithm shown in the previous section appears to be suitable for this purpose, because only finitely many degrees of freedom are used to calculate an electromagnetic field distribution if the calculation area is compact.

Note that Eqs. (12) and (20) are exact on a Yee lattice only after taking a zero cell size limit. This appears to cause no problem, but, there is an example in elementary particle physics showing that the discretized continuum theory is different from the original continuum theory [4]. In this example, a fermion in discretized quantum field theory generates nonphysical fermionic degrees of freedom. This problem is called fermion doubling. In essence, this phenomenon is caused by replacing differentials with differences as in Eqs. (9)–(10) and (17)–(19). Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

An algorithm in which differentials are not replaced with differences must be considered in order to avoid such problems.

As a result of Stokes' theorem, Faraday's and Ampére-Maxwell's laws in Eqs. (1) and (2) can be written in an integral form as

$$\frac{d}{dt} \int\_{\mathcal{S}} d\mathbf{a} \cdot \mathbf{B} = -\int\_{\mathcal{S}} d\mathbf{s} \cdot \mathbf{E},\tag{25}$$

$$\frac{d}{dt} \int\_{\mathcal{S}} d\mathbf{a} \cdot \mathbf{D} = \int\_{\mathcal{S}} d\mathbf{s} \cdot \mathbf{H} - \int\_{\mathcal{S}} d\mathbf{a} \cdot \mathbf{i},\tag{26}$$

where S is a compact and connected surface, ∂S is the boundary curve of the surface, da is a surface element normal to the surface, and ds is a line element parallel to the curve. Moreover, integrating both sides of Eq. (25) over t from t ¼ t<sup>0</sup> to t<sup>0</sup> þ Δt and those of Eq. (26) over t from t ¼ t<sup>0</sup> � Δt=2 to t<sup>0</sup> þ Δt=2 yields

$$\int\_{S} d\mathbf{a} \cdot \mathbf{B}(t\_0 + \Delta t, \mathbf{x}, \mathbf{y}, \mathbf{z}) = \int\_{S} d\mathbf{a} \cdot \mathbf{B}(t\_0, \mathbf{x}, \mathbf{y}, \mathbf{z}) - \int\_{t\_0}^{t\_0 + \Delta t} dt \int\_{\mathcal{S}} d\mathbf{s} \cdot \mathbf{E}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}), \tag{27}$$

$$\begin{split} \int\_{S} d\mathbf{a} \cdot \mathbf{D}(t\_0 + \Delta t/2, \mathbf{x}, \mathbf{y}, \mathbf{z}) &= \int\_{\mathcal{S}} d\mathbf{a} \cdot \mathbf{D}(t\_0 - \Delta t/2, \mathbf{x}, \mathbf{y}, \mathbf{z}) \\ &+ \int\_{t\_0 - \Delta t/2}^{t\_0 + \Delta t/2} dt \left[ \int\_{\mathcal{S}} d\mathbf{s} \cdot \mathbf{H}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}) - \int\_{\mathcal{S}} d\mathbf{a} \cdot \mathbf{i}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}) \right]. \end{split} \tag{28}$$

Note that no derivative is used in Eqs. (27) and (28), with the result that problems such as fermion doubling cannot occur. Our problem is how to approximate the integrals in Eqs. (27) and (28).

In general, when f is analytical in a region including ξ<sup>0</sup> � Δξ=2≤ ξ≤ξ<sup>0</sup> þ Δξ=2, the following relationship is satisfied:

$$\int\_{\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi f(\xi) = \int\_{\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi \sum\_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(\xi\_0)}{d\xi^n} (\xi - \xi\_0)^n,\tag{29}$$

and when g is analytical in a region including ξ<sup>0</sup> � Δξ=2 ≤ξ≤ξ<sup>0</sup> þ Δξ=2 and η<sup>0</sup> � Δη=2 ≤η≤η<sup>0</sup> þ Δη=2, the following relationship is satisfied:

$$\int\_{\mathfrak{n}\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi \int\_{\eta\_0-\Delta\eta/2}^{\eta\_0+\Delta\eta/2} d\eta \mathbf{g}(\xi,\eta) = \int\_{\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi \int\_{\eta\_0-\Delta\eta/2}^{\eta\_0+\Delta\eta/2} d\eta \sum\_{m\_\eta=0}^{\infty} \frac{\mathbf{1}}{m!m!} \frac{\partial^{m+n} \mathbf{g}(\xi\_0,\eta\_0)}{\partial\xi^m \partial\eta^n} \left(\xi-\xi\_0\right)^m \left(\eta-\eta\_0\right)^n \,\_2F\_1\left(\xi,\eta\_0;\frac{1}{2};\frac{1}{\eta\_0},\eta\_0+\frac{1}{2};\frac{1}{\eta\_0},\eta\_0+\frac{1}{2};\frac{1}{\eta\_0}\right) \tag{30}$$

The lowest-order approximations of Eqs. (29) and (30) are, respectively,

$$\int\_{\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi f(\xi) = f(\xi\_0)\Delta\xi + O(\Delta\xi)^3,\tag{31}$$

3. Integral form of Maxwell's equation and a correction

The FDTD method is a numerical method for solving Maxwell's equations using a computer. Any computer can have a finite number of degrees of freedom because it has a finite memory size. In contrast, an electromagnetic field in continuum space has infinitely many degrees of freedom, because the field exists at every point of the space–time continuum. Therefore, Maxwell's equations must be suitably approximated for us to be able to calculate them using a computer. The algorithm shown in the previous section appears to be suitable for this purpose, because only finitely many degrees of freedom are used to calculate an electromagnetic field distribution

Note that Eqs. (12) and (20) are exact on a Yee lattice only after taking a zero

cell size limit. This appears to cause no problem, but, there is an example in elementary particle physics showing that the discretized continuum theory is different from the original continuum theory [4]. In this example, a fermion in discretized quantum field theory generates nonphysical fermionic degrees of freedom. This problem is called fermion doubling. In essence, this phenomenon is caused by replacing differentials with differences as in Eqs. (9)–(10) and (17)–(19).

to the FDTD method

Recent Advances in Integral Equations

Flow of the FDTD algorithm.

Figure 2.

68

if the calculation area is compact.

$$\int\_{\xi\_0 - \Delta\xi/2}^{\xi\_0 + \Delta\xi/2} d\xi \int\_{\eta\_0 - \Delta\eta/2}^{\eta\_0 + \Delta\eta/2} d\eta \,\mathrm{g}(\xi, \eta) = \mathrm{g}(\xi\_0, \eta\_0) \Delta\xi \Delta\eta + \mathcal{O}\left(\left(\Delta\xi, \Delta\eta\right)^4\right). \tag{32}$$

Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup>

<sup>þ</sup> <sup>∂</sup><sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81338

� Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y0; z<sup>0</sup>

�Ex t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup>

�Ey t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup>

þEx t þ Δt=2; x0; y<sup>0</sup> � Δy=2; z<sup>0</sup> � �Δ<sup>x</sup> <sup>þ</sup>

� �ΔxΔ<sup>z</sup> <sup>þ</sup>

yDy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup>

� �ΔxΔ<sup>z</sup> <sup>þ</sup>

yDy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup>

<sup>þ</sup> Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> <sup>þ</sup>

� Hz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup>

þ Hz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup>

� �ΔxΔ<sup>z</sup> � <sup>1</sup>

ziyðt0; x1; y1; z1ÞΔxð Þ Δz

�iy t0; x1; y1; z<sup>1</sup>

� 1 <sup>24</sup> <sup>∂</sup><sup>2</sup>

relationship

71

Dy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup>

¼ Dy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup>

<sup>þ</sup> <sup>∂</sup><sup>2</sup>

<sup>þ</sup> <sup>∂</sup><sup>2</sup>

�

� �ΔxΔ<sup>y</sup> <sup>þ</sup>

¼ Bz t0; x0; y0; z<sup>0</sup>

�

� �ΔxΔ<sup>y</sup> <sup>þ</sup>

yBz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup>

1 <sup>24</sup> <sup>∂</sup><sup>2</sup>

� �Δ<sup>y</sup> <sup>þ</sup>

� �Δ<sup>x</sup> � <sup>1</sup>

� �Δ<sup>y</sup> � <sup>1</sup>

approximated by applying Eqs. (35) and (36) to yield

� �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup>

� �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup>

� �Δ<sup>z</sup> � <sup>1</sup>

� Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> � <sup>1</sup>

� �Δ<sup>z</sup> <sup>þ</sup>

24 ∂2

> 2 �

� �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup>

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

xBz t0; x0; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

> 1 <sup>24</sup> <sup>∂</sup><sup>2</sup>

<sup>24</sup> <sup>∂</sup><sup>2</sup>

<sup>24</sup> <sup>∂</sup><sup>2</sup>

1 <sup>24</sup> <sup>∂</sup><sup>2</sup>

When S is the right-hand surface of the cyan cell in Figure 1, Eq. (28) is

1 24 ∂2

h

1 24 ∂2

h

1 24 ∂2

24 ∂2

<sup>24</sup> <sup>∂</sup><sup>2</sup>

xiyðt0; x1; y1; z1Þð Þ Δx

:

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup>

There are second derivatives in Eqs. (37) and (38), but they are not calculated

in the FDTD method. Therefore, the second derivatives are determined from the calculated electromagnetic field. To determine the second derivatives, the

1 24 ∂2

i

i

1 <sup>24</sup> <sup>∂</sup><sup>2</sup>

h

xBz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

<sup>Δ</sup><sup>y</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

� �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> h i

xEx t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

yEy t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup>

xDy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup>

xDy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup>

xExð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>0; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y=2; <sup>z</sup>0Þð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

� �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

� �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

xHxðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2Þð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

� �ð Þ <sup>Δ</sup><sup>z</sup> <sup>3</sup>

xHx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup>

� �ð Þ <sup>Δ</sup><sup>z</sup> <sup>3</sup>

zHz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup>

zHz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup>

2 Δz

yEyð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>0; <sup>z</sup>0Þð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup>

i

Δy

�

Δy

Δy

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup>

(37)

(38)

:

yBz t0; x0; y0; z<sup>0</sup>

An algorithm for the FDTD method supported by the lowest-order approximation of the integral form of Maxwell's equations is derived by applying Eqs. (31) and (32) to Eqs. (27) and (28). When S is the top surface of the yellow cell in Figure 1, Eq. (27) is approximated as

$$\begin{split} B\_{\mathbf{z}}(\mathbf{t}\_{0}+\Delta t,\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{z}\_{0})\Delta\mathbf{x}\Delta\mathbf{y} &= B\_{\mathbf{z}}(\mathbf{t}\_{0},\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{z}\_{0})\Delta\mathbf{x}\Delta\mathbf{y} - \left[E\_{\mathbf{y}}(\mathbf{t}+\Delta t/2,\mathbf{x}\_{0}+\Delta\mathbf{x}/2,\mathbf{y}\_{0},\mathbf{z}\_{0})\Delta\mathbf{y}\right] \\ &- E\_{\mathbf{x}}(\mathbf{t}+\Delta t/2,\mathbf{x}\_{0},\mathbf{y}\_{0}+\Delta\mathbf{y}/2,\mathbf{z}\_{0})\Delta\mathbf{x} - E\_{\mathbf{y}}(\mathbf{t}+\Delta t/2,\mathbf{x}\_{0}-\Delta\mathbf{x}/2,\mathbf{y}\_{0},\mathbf{z}\_{0})\Delta\mathbf{y} \\ &+ E\_{\mathbf{x}}\left(\mathbf{t}+\Delta t/2,\mathbf{x}\_{0},\mathbf{y}\_{0}-\Delta\mathbf{y}/2,\mathbf{z}\_{0}\right)\Delta\mathbf{x}\right]\Delta\mathbf{t} + O(\Delta\mathbf{t})^{3}, \end{split} \tag{33}$$

where x0; y0; z<sup>0</sup> � � is the center coordinates of the surface. Comparison of Eq. (12) with Eq. (33) reveals that they are essentially the same.

When S is the right surface of the cyan cell in Figure 1, Eq. (28) can be approximated as

$$\begin{aligned} D\_{\mathcal{V}} \left( t\_{0} + \Delta t / 2, \mathbf{x}\_{1}, \mathbf{y}\_{1}, \mathbf{z}\_{1} \right) \Delta \mathbf{x} \Delta \mathbf{z} &= D\_{\mathcal{V}} \left( t\_{0} - \Delta t / 2, \mathbf{x}\_{1}, \mathbf{y}\_{1}, \mathbf{z}\_{1} \right) \Delta \mathbf{x} \Delta \mathbf{z} + \left[ H\_{\mathbf{x}} \left( t\_{0}, \mathbf{x}\_{1}, \mathbf{y}\_{1}, \mathbf{z}\_{1} + \Delta \mathbf{z} / 2 \right) \Delta \mathbf{x} \right. \\ &- H\_{\mathbf{z}} \left( t\_{0}, \mathbf{x}\_{1} + \Delta \mathbf{x} / 2, \mathbf{y}\_{1}, \mathbf{z}\_{1} \right) \Delta \mathbf{x} - H\_{\mathbf{x}} \left( t\_{0}, \mathbf{x}\_{1}, \mathbf{y}\_{1}, \mathbf{z}\_{1} - \Delta \mathbf{z} / 2 \right) \Delta \mathbf{x} + H\_{\mathbf{z}} \left( t\_{0}, \mathbf{x}\_{1} - \Delta \mathbf{x} / 2, \mathbf{y}\_{1}, \mathbf{z}\_{1} \right) \Delta \mathbf{x} \\ &- i\_{\mathcal{V}} \left( t\_{0}, \mathbf{x}\_{1}, \mathbf{y}\_{1}, \mathbf{z}\_{1} \right) \Delta \mathbf{x} \Delta \mathbf{x} \Big] \Delta \mathbf{t} + O(\Delta t)^{3}, \tag{34} \tag{35} \end{aligned} \tag{36}$$

where x1; y1; z<sup>1</sup> � � is the center coordinates of the surface. Comparison of Eq. (20) with Eq. (34) reveals that they are essentially the same. Therefore, the original FDTD method, which is based on the differential form of Maxwell's equations, is the same as the FDTD method supported by the lowest-order approximation of the integral form of those equations.

Next, an algorithm for the FDTD method supported by the next-to-the-lowestorder approximation of the integral form of Maxwell's equation is derived. In this case, the next-to-the-lowest-order approximation is applied only in the spatial directions, and the lowest-order approximation is applied in the time direction. Hereafter, the FDTD method supported by the next-to-the-lowest-order approximation of integrals is called the corrected FDTD method.

In general, the next-to-the-lowest-order approximations of Eqs. (29) and (30) are, respectively,

$$\int\_{\xi\_0-\Delta\xi/2}^{\xi\_0+\Delta\xi/2} d\xi f(\xi) = f(\xi\_0)\Delta\xi + \frac{1}{24} \frac{d^2 f(\xi\_0)}{d\xi^2} (\Delta\xi)^3 + O(\Delta\xi)^5,\tag{35}$$

$$\begin{split} \int\_{\xi\_{0}-\Delta\xi/2}^{\xi\_{0}+\Delta\xi/2} d\xi \int\_{\eta\_{0}-\Delta\eta/2}^{\eta\_{0}+\Delta\eta/2} d\eta \mathbf{g}(\xi,\eta) &= \mathbf{g}(\xi\_{0},\eta\_{0})\Delta\xi \Delta\eta + \frac{1}{24} \left( \frac{\partial^{2}\mathbf{g}(\xi,\eta)}{\partial\xi^{2}} (\Delta\xi)^{3} \Delta\eta + \frac{\partial^{2}\mathbf{g}(\xi,\eta)}{\partial\eta^{2}} \Delta\xi (\Delta\eta)^{3} \right) \\ &+ O(\Delta\xi,\Delta\eta)^{6}. \end{split} \tag{36}$$

When S is the top surface of the yellow cell in Figure 1, Eq. (27) is approximated by applying Eqs. (35) and (36) to yield

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup> � �ΔxΔ<sup>y</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>∂</sup><sup>2</sup> xBz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> Δy h <sup>þ</sup> <sup>∂</sup><sup>2</sup> yBz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup> � �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> i ¼ Bz t0; x0; y0; z<sup>0</sup> � �ΔxΔ<sup>y</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>∂</sup><sup>2</sup> xBz t0; x0; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> <sup>Δ</sup><sup>y</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> yBz t0; x0; y0; z<sup>0</sup> � �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> h i � Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y0; z<sup>0</sup> � �Δ<sup>y</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>∂</sup><sup>2</sup> yEyð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>0; <sup>z</sup>0Þð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> � �Ex t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup> � �Δ<sup>x</sup> � <sup>1</sup> <sup>24</sup> <sup>∂</sup><sup>2</sup> xEx t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> �Ey t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup> � �Δ<sup>y</sup> � <sup>1</sup> <sup>24</sup> <sup>∂</sup><sup>2</sup> yEy t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup> � �ð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> þEx t þ Δt=2; x0; y<sup>0</sup> � Δy=2; z<sup>0</sup> � �Δ<sup>x</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>∂</sup><sup>2</sup> xExð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>0; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y=2; <sup>z</sup>0Þð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> � <sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (37)

When S is the right-hand surface of the cyan cell in Figure 1, Eq. (28) is approximated by applying Eqs. (35) and (36) to yield

Dy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup> � �ΔxΔ<sup>z</sup> <sup>þ</sup> 1 24 ∂2 xDy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> Δy h <sup>þ</sup> <sup>∂</sup><sup>2</sup> yDy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup> � �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> i ¼ Dy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup> � �ΔxΔ<sup>z</sup> <sup>þ</sup> 1 24 ∂2 xDy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> Δy h <sup>þ</sup> <sup>∂</sup><sup>2</sup> yDy t<sup>0</sup> � Δt=2; x1; y1; z<sup>1</sup> � �Δxð Þ <sup>Δ</sup><sup>y</sup> <sup>3</sup> i <sup>þ</sup> Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> <sup>þ</sup> 1 24 ∂2 xHxðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2Þð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> � � Hz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup> � �Δ<sup>z</sup> � <sup>1</sup> 24 ∂2 zHz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup> � �ð Þ <sup>Δ</sup><sup>z</sup> <sup>3</sup> � Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> � <sup>1</sup> <sup>24</sup> <sup>∂</sup><sup>2</sup> xHx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> � �ð Þ <sup>Δ</sup><sup>x</sup> <sup>3</sup> þ Hz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup> � �Δ<sup>z</sup> <sup>þ</sup> 1 24 ∂2 zHz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup> � �ð Þ <sup>Δ</sup><sup>z</sup> <sup>3</sup> �iy t0; x1; y1; z<sup>1</sup> � �ΔxΔ<sup>z</sup> � <sup>1</sup> 24 ∂2 xiyðt0; x1; y1; z1Þð Þ Δx 2 Δz � 1 <sup>24</sup> <sup>∂</sup><sup>2</sup> ziyðt0; x1; y1; z1ÞΔxð Þ Δz 2 � <sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (38)

There are second derivatives in Eqs. (37) and (38), but they are not calculated in the FDTD method. Therefore, the second derivatives are determined from the calculated electromagnetic field. To determine the second derivatives, the relationship

Z ξ0þΔξ=2

dξ

Recent Advances in Integral Equations

Z<sup>η</sup>0þΔη=<sup>2</sup>

dη gð Þ¼ ξ; η g ξ0; η<sup>0</sup> ð ÞΔξΔη þ O ð Þ Δξ; Δη

� �ΔxΔ<sup>y</sup> � Ey <sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup>

� �ΔxΔ<sup>z</sup> <sup>þ</sup> Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> �

<sup>d</sup>ξ<sup>2</sup> ð Þ <sup>Δ</sup><sup>ξ</sup> <sup>3</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>ξ</sup> <sup>5</sup>

Δη þ ∂2 gð Þ ξ; η <sup>∂</sup>η<sup>2</sup> <sup>Δ</sup>ξð Þ <sup>Δ</sup><sup>η</sup>

� �

,

� �Δ<sup>x</sup> � Ey <sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x=2; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

�

An algorithm for the FDTD method supported by the lowest-order approximation of the integral form of Maxwell's equations is derived by applying Eqs. (31) and (32) to Eqs. (27) and (28). When S is the top surface of the yellow cell in Figure 1,

� � is the center coordinates of the surface. Comparison of

When S is the right surface of the cyan cell in Figure 1, Eq. (28) can be approx-

� �Δ<sup>z</sup> � Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> � �Δ<sup>x</sup> <sup>þ</sup> Hz <sup>t</sup>0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup>

� � is the center coordinates of the surface. Comparison of Eq. (20)

<sup>4</sup> � �

� �Δy �

� �Δy

� �Δz

, (35)

3

(36)

: (32)

(33)

(34)

η0�Δη=2

� �ΔxΔ<sup>y</sup> <sup>¼</sup> Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

� Ex t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup>

þ Ex t þ Δt=2; x0; y<sup>0</sup> � Δy=2; z<sup>0</sup> � �Δx

Eq. (12) with Eq. (33) reveals that they are essentially the same.

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup>

mation of integrals is called the corrected FDTD method.

dξ fð Þ¼ ξ f ξ<sup>0</sup> ð ÞΔξ þ

dηgð Þ¼ ξ; η g ξ0; η<sup>0</sup> ð ÞΔξΔη þ

þ Oð Þ Δξ; Δη

,

with Eq. (34) reveals that they are essentially the same. Therefore, the original FDTD method, which is based on the differential form of Maxwell's equations, is the same as the FDTD method supported by the lowest-order approximation of the

Next, an algorithm for the FDTD method supported by the next-to-the-lowestorder approximation of the integral form of Maxwell's equation is derived. In this case, the next-to-the-lowest-order approximation is applied only in the spatial directions, and the lowest-order approximation is applied in the time direction. Hereafter, the FDTD method supported by the next-to-the-lowest-order approxi-

In general, the next-to-the-lowest-order approximations of Eqs. (29) and (30)

1 24 d2 f ξ<sup>0</sup> ð Þ

6 :

When S is the top surface of the yellow cell in Figure 1, Eq. (27) is approximated

1 24 ∂2 gð Þ ξ; η <sup>∂</sup>ξ<sup>2</sup> ð Þ <sup>Δ</sup><sup>ξ</sup> <sup>3</sup>

� �ΔxΔ<sup>z</sup> <sup>¼</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup>

�

ξ0�Δξ=2

Eq. (27) is approximated as

Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup>

where x0; y0; z<sup>0</sup>

Dy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup>

�iy t0; x1; y1; z<sup>1</sup> � �ΔxΔz

where x1; y1; z<sup>1</sup>

are, respectively,

dξ

ξ0Z þΔξ=2

ξ0�Δξ=2

70

� Hz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup>

integral form of those equations.

Z ξ0þΔξ=2

ξ0�Δξ=2

Z<sup>η</sup>0þΔη=<sup>2</sup>

η0�Δη=2

by applying Eqs. (35) and (36) to yield

imated as

$$f(\xi\_0 + \Delta \xi) + f(\xi\_0 - \Delta \xi) = \mathcal{Y}(\xi) + \frac{d^2 f(\xi\_0)}{d\xi^2} \left(\Delta \xi\right)^2 + O\left(\Delta \xi\right)^4,\tag{39}$$

Note that points xi � Δx; yi

; zi

DOI: http://dx.doi.org/10.5772/intechopen.81338

cell including xi; yi

using these equations.

σ ξ<sup>ð</sup> ; <sup>η</sup>; <sup>ξ</sup> <sup>þ</sup> <sup>m</sup>Δξ; <sup>η</sup> <sup>þ</sup> <sup>n</sup>ΔηÞ ¼ <sup>5</sup>

∑ m, n

where

∑ m, n

∑ p, <sup>q</sup>

� <sup>11</sup> 12

þ 1 24 ��

<sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 � �Δy; <sup>z</sup>0<sup>Þ</sup>

73

�

Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup>

; zi � � and xi; yi

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

The LHSs of Eqs. (40) and (41) can be written symbolically as

<sup>6</sup> <sup>δ</sup>m,0δn,<sup>0</sup> <sup>þ</sup>

and δp, <sup>q</sup> is the Kronecker delta defined as

The inverse operator "σ�<sup>1</sup>" is defined as

Using σ�<sup>1</sup> enables Eq. (40) to be rewritten as

1 2

1 2

2

� �ΔxΔ<sup>y</sup> � <sup>∑</sup>

1 2

σ ξð Þ ; <sup>η</sup>; <sup>ξ</sup> <sup>þ</sup> <sup>p</sup>Δξ; <sup>η</sup> <sup>þ</sup> <sup>q</sup>Δ<sup>η</sup> <sup>σ</sup>�<sup>1</sup>

� �ΔxΔ<sup>y</sup> <sup>¼</sup> Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ m þ

� Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ mΔx; y<sup>0</sup> þ n þ

Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ m þ

� Exðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ ð Þ m þ 1 Δx; y<sup>0</sup> þ n þ

�Exðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ ð Þ m � 1 Δx; y<sup>0</sup> þ n þ

� Δx � Δt � <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> :

� Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

<sup>σ</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δ<sup>y</sup> � �Bz <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δy; <sup>z</sup><sup>0</sup>

1

<sup>δ</sup>p,q <sup>¼</sup> <sup>1</sup> ð Þ <sup>p</sup> <sup>¼</sup> <sup>q</sup>

m, <sup>n</sup>

� �

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

1 2

1 2

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δy; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0Þ þ Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup>

!

� �Δy; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

�� �

�

0 ð Þ p 6¼ q

<sup>σ</sup> <sup>x</sup>1; <sup>z</sup>1; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>Δ<sup>z</sup> � �Dy <sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>Δ<sup>z</sup> � �, (43)

; zi � <sup>Δ</sup><sup>z</sup> � � are at adjacent cells to the

� �, (42)

<sup>24</sup> ð Þ <sup>δ</sup>m, <sup>1</sup>δn,<sup>0</sup> <sup>þ</sup> <sup>δ</sup>m,�<sup>1</sup>δn,<sup>0</sup> <sup>þ</sup> <sup>δ</sup>m,0δn, <sup>1</sup> <sup>þ</sup> <sup>δ</sup>m,0δn,�<sup>1</sup> ,

ðξ þ pΔξ; η þ qΔη; ξ þ mΔξ; η þ nΔηÞ ¼ δm,0δn,0:

<sup>σ</sup>�<sup>1</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δ<sup>y</sup> � �

� �Δy; <sup>z</sup>0Þ þ Exðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

� �Δy; <sup>z</sup>0Þ � Exðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup><sup>0</sup>

2

2

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δy; <sup>z</sup>0<sup>Þ</sup>

2 � �Δy; <sup>z</sup>0<sup>Þ</sup>

> 1 2

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup>

: (45)

(44)

(46)

Δy

� Δy

(47)

Δx �

2 � �Δy; <sup>z</sup><sup>0</sup>

� �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup>

� �. Therefore, all terms in Eqs. (40) and (41) are fields defined

on the Yee lattice. However, in contrast to the original FDTD method, the left-hand sides (LHSs) of these equations are a linear combination of fields at five points. Therefore, it is impossible to directly determine the values of fields at new times

for any function f is applied. Applying Eq. (39) to Eq. (37) yields

5 <sup>6</sup> Bz <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup> <sup>þ</sup> 1 24 Bz t<sup>0</sup> þ Δt; x<sup>0</sup> þ Δx; y0; z<sup>0</sup> <sup>þ</sup> Bzðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup>0; <sup>z</sup>0<sup>Þ</sup> þ Bz t<sup>0</sup> þ Δt; x0; y<sup>0</sup> þ Δy; z<sup>0</sup> <sup>þ</sup> Bzðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup> xΔy <sup>¼</sup> <sup>5</sup> <sup>6</sup> Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup> <sup>þ</sup> 1 24 Bz t0; x<sup>0</sup> þ Δx; y0; z<sup>0</sup> <sup>þ</sup> Bzðt0; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup>0; <sup>z</sup>0<sup>Þ</sup> þ Bzðt0; x0; y<sup>0</sup> þ Δy; z0Þ þ Bzðt0; x0; y<sup>0</sup> � Δy; z0Þ ΔxΔy � <sup>11</sup> 12 Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y0; z<sup>0</sup> <sup>þ</sup> 1 24 Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y<sup>0</sup> þ Δy; z<sup>0</sup> þ Eyðt þ Δt=2; x<sup>0</sup> þ Δx=2; y<sup>0</sup> � Δy; z0Þ <sup>Δ</sup><sup>y</sup> � <sup>11</sup> 12 Ex t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup> þ 1 24 Ex t þ Δt=2; x<sup>0</sup> þ Δx; y<sup>0</sup> þ Δy=2; z<sup>0</sup> <sup>þ</sup> Exð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>y=2; <sup>z</sup>0<sup>Þ</sup> Δx � <sup>11</sup> 12 Ey t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>ð</sup>Eyð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x=2; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup> <sup>þ</sup> Eyð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x=2; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y; <sup>z</sup>0ÞÞ Δy þ 11 12 Ex t þ Δt=2; x0; y<sup>0</sup> � Δy=2; z<sup>0</sup> þ 1 24 Ex t þ Δt=2; x<sup>0</sup> þ Δx; y<sup>0</sup> � Δy=2; z<sup>0</sup> <sup>þ</sup> Exð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y=2; <sup>z</sup>0<sup>Þ</sup> Δx Δt <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (40)

Applying Eq. (39) to Eq. (38) yields

5 <sup>6</sup> Dy <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>ð</sup>Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup> <sup>þ</sup> Dy <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>zÞÞ ΔxΔz <sup>¼</sup> <sup>5</sup> <sup>6</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>ð</sup>Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup> <sup>þ</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>zÞÞ ΔxΔz þ 11 12 Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup> 1 <sup>24</sup> <sup>ð</sup>Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2Þ þ Hxðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2ÞÞ Δx � <sup>11</sup> 12 Hz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup> <sup>þ</sup> 1 24 Hz <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z<sup>Þ</sup> Δz � <sup>11</sup> 12 Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup> 1 24 Hx <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup> Hxðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=2<sup>Þ</sup> Δx þ 11 12 Hz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup> <sup>þ</sup> 1 24 Hz <sup>t</sup>0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Hzðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z<sup>Þ</sup> Δz � 5 <sup>6</sup> iy <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> 1 <sup>24</sup> iy <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> iyðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ iyðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z<sup>Þ</sup> <sup>þ</sup>iyðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>zÞÞ ΔxΔz <sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (41)

### Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

Note that points xi � Δx; yi ; zi � � and xi; yi ; zi � <sup>Δ</sup><sup>z</sup> � � are at adjacent cells to the cell including xi; yi ; zi � �. Therefore, all terms in Eqs. (40) and (41) are fields defined on the Yee lattice. However, in contrast to the original FDTD method, the left-hand sides (LHSs) of these equations are a linear combination of fields at five points. Therefore, it is impossible to directly determine the values of fields at new times using these equations.

The LHSs of Eqs. (40) and (41) can be written symbolically as

$$\sum\_{m\_0 n} \sigma(\mathbf{x}\_0, \mathbf{y}\_0; \mathbf{x}\_0 + m\Delta \mathbf{x}, \mathbf{y}\_0 + n\Delta \mathbf{y}) B\_\mathbf{z} (t\_0 + \Delta t, \mathbf{x}\_0 + m\Delta \mathbf{x}, \mathbf{y}\_0 + n\Delta \mathbf{y}, \mathbf{z}\_0),\tag{42}$$

$$\sum\_{m\_{\mathcal{V}}n} \sigma(\mathbf{x}\_1, \mathbf{z}\_1; \mathbf{x}\_1 + m\Delta x, \mathbf{y}\_1, \mathbf{z}\_1 + n\Delta z) D\_{\mathcal{V}} (t + \Delta t/2, \mathbf{x}\_1 + m\Delta x, \mathbf{y}\_1, \mathbf{z}\_1 + n\Delta z), \quad \text{(43)}$$

where

<sup>f</sup>ð Þþ <sup>ξ</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup><sup>ξ</sup> <sup>f</sup>ð Þ¼ <sup>ξ</sup><sup>0</sup> � <sup>Δ</sup><sup>ξ</sup> <sup>2</sup>fð Þþ <sup>ξ</sup> <sup>d</sup><sup>2</sup>

1 24

> 1 24

þ Bzðt0; x0; y<sup>0</sup> þ Δy; z0Þ þ Bzðt0; x0; y<sup>0</sup> � Δy; z0Þ

Ex t þ Δt=2; x<sup>0</sup> þ Δx; y<sup>0</sup> þ Δy=2; z<sup>0</sup>

Ex t þ Δt=2; x<sup>0</sup> þ Δx; y<sup>0</sup> � Δy=2; z<sup>0</sup>

Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y0; z<sup>0</sup> <sup>þ</sup>

þ Eyðt þ Δt=2; x<sup>0</sup> þ Δx=2; y<sup>0</sup> � Δy; z0Þ

Ey t þ Δt=2; x<sup>0</sup> � Δx=2; y0; z<sup>0</sup> <sup>þ</sup>

þ Eyðt þ Δt=2; x<sup>0</sup> � Δx=2; y<sup>0</sup> � Δy; z0ÞÞ

Applying Eq. (39) to Eq. (38) yields

Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup>

Hz t0; x<sup>1</sup> þ Δx=2; y1; z<sup>1</sup> <sup>þ</sup>

Hz t0; x<sup>1</sup> � Δx=2; y1; z<sup>1</sup> <sup>þ</sup>

<sup>6</sup> iy <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup>

þiyðt0; x1; y1; z<sup>1</sup> � ΔzÞÞ

Hx <sup>t</sup>0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup>

1

 ΔxΔz 

1

<sup>þ</sup> Dy <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>zÞÞ

<sup>þ</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>zÞÞ

1

1 24

1 24

1 24

<sup>24</sup> iy <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup>

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup>

1

þ Bz t<sup>0</sup> þ Δt; x0; y<sup>0</sup> þ Δy; z<sup>0</sup>

Recent Advances in Integral Equations

<sup>6</sup> Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup> <sup>þ</sup>

5

<sup>6</sup> Bz <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup> <sup>þ</sup>

> � <sup>11</sup> 12

þ 1 24

þ 1 24

5

<sup>¼</sup> <sup>5</sup>

þ 11 12

� <sup>11</sup> 12

� <sup>11</sup> 12

þ 11 12

72

� 5

<sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> :

<sup>6</sup> Dy <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup>

<sup>6</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup>

� <sup>11</sup> 12

<sup>¼</sup> <sup>5</sup>

for any function f is applied. Applying Eq. (39) to Eq. (37) yields

<sup>þ</sup> Bzðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup>

Bz t<sup>0</sup> þ Δt; x<sup>0</sup> þ Δx; y0; z<sup>0</sup> <sup>þ</sup> Bzðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup>0; <sup>z</sup>0<sup>Þ</sup>

Bz t0; x<sup>0</sup> þ Δx; y0; z<sup>0</sup> <sup>þ</sup> Bzðt0; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup>0; <sup>z</sup>0<sup>Þ</sup>

f ξ<sup>0</sup> ð Þ

 ΔxΔy

1 24

<sup>þ</sup> Exð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>y=2; <sup>z</sup>0<sup>Þ</sup>

<sup>þ</sup> Exð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> � <sup>Δ</sup>y=2; <sup>z</sup>0<sup>Þ</sup>

11 12

<sup>24</sup> <sup>ð</sup>Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ Dyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup>

<sup>24</sup> <sup>ð</sup>Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ Dyðt<sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup>

Hz <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z<sup>Þ</sup>

Hx <sup>t</sup>0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=<sup>2</sup> <sup>þ</sup> Hxðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z=2<sup>Þ</sup>

Hz <sup>t</sup>0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>z</sup> <sup>þ</sup> Hzðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x=2; <sup>y</sup>1; <sup>z</sup><sup>1</sup> � <sup>Δ</sup>z<sup>Þ</sup>

<sup>þ</sup> iyðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup>1Þ þ iyðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z<sup>Þ</sup>

:

 ΔxΔz

 ΔxΔz

<sup>24</sup> <sup>ð</sup>Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2Þ þ Hxðt0; <sup>x</sup><sup>1</sup> � <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>Δ</sup>z=2ÞÞ

 <sup>Δ</sup><sup>y</sup> � <sup>11</sup> 12

1

 Δy þ <sup>d</sup>ξ<sup>2</sup> ð Þ <sup>Δ</sup><sup>ξ</sup> <sup>2</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>ξ</sup> <sup>4</sup>

 xΔy

Ey t þ Δt=2; x<sup>0</sup> þ Δx=2; y<sup>0</sup> þ Δy; z<sup>0</sup>

<sup>24</sup> <sup>ð</sup>Eyð<sup>t</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> � <sup>Δ</sup>x=2; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup>

Ex t þ Δt=2; x0; y<sup>0</sup> þ Δy=2; z<sup>0</sup> 

Ex t þ Δt=2; x0; y<sup>0</sup> � Δy=2; z<sup>0</sup> 

, (39)

Δx

Δx Δt

(40)

Δx

Δz

Δx

Δz

(41)

$$\sigma(\xi,\eta;\xi+m\Delta\xi,\eta+n\Delta\eta) = \frac{5}{6}\delta\_{m,0}\delta\_{n,0} + \frac{1}{24}(\delta\_{m,1}\delta\_{n,0} + \delta\_{m,-1}\delta\_{n,0} + \delta\_{m,0}\delta\_{n,1} + \delta\_{m,0}\delta\_{n,-1}),\tag{44}$$

and δp, <sup>q</sup> is the Kronecker delta defined as

$$\delta\_{p,q} = \begin{cases} 1 & (p=q) \\ 0 & (p \neq q) \end{cases} \tag{45}$$

The inverse operator "σ�<sup>1</sup>" is defined as

$$\sum\_{p,q} \sigma(\xi,\eta;\xi+p\Delta\xi,\eta+q\Delta\eta)\sigma^{-1}(\xi+p\Delta\xi,\eta+q\Delta\eta;\xi+m\Delta\xi,\eta+n\Delta\eta) = \delta\_{m,0}\delta\_{n,0}.\tag{46}$$

Using σ�<sup>1</sup> enables Eq. (40) to be rewritten as

Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup> � �ΔxΔ<sup>y</sup> <sup>¼</sup> Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup> � �ΔxΔ<sup>y</sup> � <sup>∑</sup> m, <sup>n</sup> <sup>σ</sup>�<sup>1</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δ<sup>y</sup> � � � <sup>11</sup> 12 Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ m þ 1 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δy; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>Δy; <sup>z</sup>0<sup>Þ</sup> �� � Δy � � Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ mΔx; y<sup>0</sup> þ n þ 1 2 � �Δy; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup>Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 � �Δy; <sup>z</sup>0<sup>Þ</sup> � � Δx � þ 1 24 Eyðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ m þ 1 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0Þ þ Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> 1 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup> �� � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0Þ � Eyðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 � �Δx; <sup>y</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup>y; <sup>z</sup>0<sup>Þ</sup> � Δy � Exðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ ð Þ m þ 1 Δx; y<sup>0</sup> þ n þ 1 2 � �Δy; <sup>z</sup>0Þ þ Exðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 � �Δy; <sup>z</sup><sup>0</sup> ! �Exðt<sup>0</sup> þ Δt=2; x<sup>0</sup> þ ð Þ m � 1 Δx; y<sup>0</sup> þ n þ 1 2 � �Δy; <sup>z</sup>0Þ � Exðt<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t=2; <sup>x</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 � �Δy; <sup>z</sup>0<sup>Þ</sup> � Δx � Δt � <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (47)

In addition, Eq. (40) can also be rewritten as

Dy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup> <sup>Δ</sup>xΔ<sup>z</sup> <sup>¼</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> iyðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup> <sup>Δ</sup>xΔ<sup>z</sup> þ ∑ m, <sup>n</sup> σ�<sup>1</sup> ð Þ x1; z1; x<sup>1</sup> þ mΔx; z<sup>1</sup> þ nΔz � <sup>11</sup> 12 Hxðt0; x<sup>1</sup> þ mΔx; y1; z<sup>1</sup> þ n þ 1 2 <sup>Δ</sup>zÞ � Hxðt0; <sup>x</sup>1<sup>þ</sup> mΔx; y <sup>1</sup>; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 <sup>Δ</sup>z<sup>Þ</sup> Δx � Hzðt0; x<sup>1</sup> þ m þ 1 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>ΔzÞ � Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>Δz<sup>Þ</sup> Δz þ 1 24 Hxðt0; x<sup>1</sup> þ ð Þ m þ 1 Δx; y1; z<sup>1</sup> þ n þ 1 2 <sup>Δ</sup>zÞ þ Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> <sup>þ</sup> 1 2 <sup>Δ</sup><sup>z</sup> � Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 <sup>Δ</sup>zÞ � Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> 2 <sup>Δ</sup>z<sup>Þ</sup> Δx � Hzðt0; x<sup>1</sup> þ m þ 1 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>zÞ þ Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> 1 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup><sup>z</sup> � Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>zÞ � Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup> 2 <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>Δ</sup>zÞÞΔ<sup>z</sup> Δ<sup>t</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> : (48)

Then, the algorithm of the corrected FDTD method, which is supported by the next-to-the-lowest-order approximation, can be obtained by using Eqs. (48) and (47) repeatedly.
