Fast High-Order Algorithms for Electromagnetic Scattering Problem from Finite Array of Cavities in TE Case with High Wave Numbers

Abstract

In this paper, fast high-order finite difference algorithms for solving the electromagnetic scattering from the finite array of two-dimensional rectangular cavities are proposed in TE polarization. The scattering problem from the cavity array is described as coupled Helmholtz equations with transparent boundary conditions on open apertures. Second-order and fourth-order schemes for solving the coupled systems are developed in TE polarization respectively. A special technique is applied to construct a fourth-order scheme for transparent boundary conditions. Further, we propose fast algorithms which can simplify the larger global system to a small linear interface system on the apertures of cavities. Numerical experiments show the validity and efficiency of the proposed fast algorithms for solving the scattering problem with high wave numbers.

1. Introduction

Electromagnetic scattering from the finite array of large cavities has attracted much attention in industry and the military. The cavity structures are very common, such as jet engine inlet ducts, exhaust nozzles, and cavity-backed antennas. The radar cross section (RCS) is one of the physical parameters of interest, which measures the detectability of a target by a radar system. The accurate prediction of the RCS of the cavities is significant due to its dominance of the overall RCS of the targets.

Numerous works have been devoted to solving the scattering problems from cavities, such as the method of moment (MoM) [1,2], finite element-boundary integral (FE-BI) method [3,4,5,6,7,8], the finite difference method (FDM) [9,10,11,12,13,14], and other methods [15,16,17,18,19,20,21,22,23,24]. The cavity scattering problems arouse a wide discussion in computational mathematics. Ammari et al. [25,26] gave the result of the existence and uniqueness of the solution to the cavity scattering problem. More mathematical analysis for the cavity scattering problem can be found in [10,25,27].

Most of the studies above were used to solve the electromagnetic scattering from the single cavity model, which limits the practical applications of the model problem in industry and the military. More recently, the finite array of cavity scattering problem was examined in [8,28,29]. Alavikia and Ramahi [8] presented a hybrid finite element boundary integral algorithm to solve the problem of scattering from a finite array of cavities. Li and Wood [28] applied the finite element method to solve the electromagnetic scattering from multiple cavities. Wu and Zheng [29] proposed a perfectly matched layer (PML) method for solving the electromagnetic scattering from multiple cavities. Unfortunately, when the wave number $k_0$ is high, or the size of the cavity is large compared to the wavelength of the incident wave, the scale of the system will become larger, the computational cost will increase, and the condition number of the matrix will become worse. This has been a crucial topic in the scattering problem for large cavities. A straightforward change of coordinates yields the equivalence of high wave numbers and large cavity problems. For the electromagnetic scattering with high wave numbers, Bao and Sun [10] proposed a fast algorithm for solving the electromagnetic scattering from a large rectangular open cavity in TM polarization.

This paper mainly focuses on the scattering from a finite array of two-dimensional rectangular cavities in TE polarization. In the existing literature or research, the electromagnetic scattering problems in TM polarization have been studied in detail while those in the TE case are seldom discussed. In the TE case, the electric field is perpendicular to the invariant direction. The main difficulty of the scattering from the array of cavities is the complex interaction boundary conditions. In this paper, firstly, by introducing a new transparent boundary condition on the cavity apertures, the finite array of cavity scattering problem is reduced to a boundary value problem of the two-dimensional Helmholtz equation. Considering the boundary coupling of the finite array of cavities, we make extensions of the total field on the apertures. We develop the compact second and fourth-order finite difference schemes respectively to discretize the reduced Helmholtz equation in TE polarization. Moreover, the Neumann boundary condition on the boundaries and the transparent boundary conditions on the apertures need a special approximation. In particular, in order to deal with the problem with high wave numbers, we introduce the discrete Fourier transformation in the horizontal (x-axis) and a Gaussian elimination in the vertical direction (y-axis) on Cartesian coordinates, which can simplify the cavity scattering problem to a small aperture system. Thus the cost of calculating the radar cross section is $O(NMlog(nM))$ with $M\times N$ meshes in each cavity, where n is the number of separate cavities.

The rest of the paper is organized as follows. In Section 2, we illustrate the model problem for the scattering from a single cavity in TE polarization. In Section 3, the compact second-order and fourth-order finite difference schemes are applied to discretize the magnetic field equation in TE polarization. In Section 4, we propose a fast algorithm for the electromagnetic scattering from the finite array of cavities with high wave numbers in TE polarization. Numerical results demonstrate the accuracy and efficiency of the proposed method in Section 5.

2. The Single Cavity Scattering Problem in TE Polarization

We consider an electromagnetic plane wave on a cavity embedded in an infinite ground plane. The total electric field and the magnetic field, denoted by $\mathbf{E}$ and $\mathbf{H}$ separately, satisfy the following time-harmonic Maxwell’s equations

$$\begin{array}{cc}& \nabla \times \mathbf{E}=\mathrm{i}\omega \mathbf{B},\\ & \nabla \times \mathbf{H}=-\mathrm{i}\omega \mathbf{D},\end{array}$$

and the constitutive relationship

$$\begin{array}{cc}& \mathbf{D}={\epsilon }_0{\epsilon }_r\mathbf{E},\\ & \mathbf{B}={\mu }_0{\mu }_r\mathbf{H},\end{array}$$

where ${\epsilon }_0=8.85\times {10}^{12}$ Fm${}^{-1}$, ${\mu }_0=4\pi \times {10}^{-7}$ Hm${}^{-1}$, ${\epsilon }_r, {\mu }_r$ are the relative electric permittivity and magnetic permeability, and $\mathrm{i}=\sqrt{-1}$ and $\omega$ is the angular frequency.

For the electromagnetic scattering problem in TE polarization, the electric field is transverse to the z-axis. The magnetic field $\mathbf{H}$ is in the form of $\mathbf{H}=(0,0,H_z)$. For ease of notation, we denote $H_z$ by u. Let the incident field take the form of $u^i=e^{\mathrm{i}(\alpha x+\beta y)}$, the scattered magnetic field can be expressed by

$$u^s=u-u^i-u^r,$$

(1)

where $u^r=e^{\mathrm{i}(\alpha x-\beta y)}$, $\alpha =k_{0}sin\theta , \beta =k_{0}cos\theta$, $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$ is the angle of the incidence, $k_0$ is the wave number in free space.

We consider the cavity filled with a homogeneous medium. The geometry of the cavity is shown in Figure 1. We assume the ground plane ${\mathsf{\Gamma }}^c$ and the wall S of the cavity are perfect electric conductors (PEC). Denote the upper half space as ${\mathbb{R}}_{+}^2=\{(x,y)\in R^2,y>0\}$. There exists a homogeneous medium of ${\epsilon }_0$ and ${\mu }_0$ in ${\mathbb{R}}_{+}^2$. Therefore, the total magnetic field in the z-axis satisfies the following equations

$$\begin{array}{cccc}\hfill & \frac{1}{{\epsilon }_r}\Delta u+k_0^2{\mu }_{r}u=0,\hfill & \hfill \mathrm{in}& \mathsf{\Omega }\cup {\mathbb{R}}_{+}^2,\hfill \\ \hfill & \frac{1}{{\epsilon }_r}\frac{\partial u}{\partial n}=0,\hfill & \hfill \mathrm{on}& (\partial \mathsf{\Omega }\backslash \mathsf{\Gamma })\cup {\mathsf{\Gamma }}^c,\hfill \end{array}$$

(2)

Figure 1. Single cavity scattering geometry.

Together with the Sommerfeld radiation boundary condition

$$\underset{r⟶\infty }{lim}\sqrt{r}\left(\frac{\partial u^s}{\partial r}-\mathrm{i}{k}_0{u}^s\right)=0,$$

(3)

where ${\epsilon }_r={\mu }_r=1$ in $R_{+}^2$.

According to Equations (1)–(3), the scattered field $u^s$ satisfies

$$\begin{array}{ccc}& & \Delta u^s+k_0^2{u}^s=0,(x,y)\in {\mathbb{R}}_{+}^2,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \frac{\partial u^s}{\partial n}=0,\mathrm{on}{\mathsf{\Gamma }}^c,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & \frac{\partial u^s}{\partial n}=\frac{\partial u}{\partial n},\mathrm{on}\mathsf{\Gamma }.\hfill \end{array}$$

(6)

In order to convert the problem (2) and (3) into the bounded domain, we adopt the upper half-space Neumann Green function for the Helmholtz equation

$$G_N\left(\mathit{x},{\mathit{x}}^{\prime }\right)=\frac{\mathrm{i}}{4}\left({\mathrm{H}}_0^{(1)}\left(k_{0}r\right)+{\mathrm{H}}_0^{(1)}\left(k_0\overline{r}\right)\right),$$

(7)

where $\mathit{x}=(x,y)$, ${\mathit{x}}^{\prime }=\left(x^{\prime },y^{\prime }\right)$, $r=\mid \mathit{x}-{\mathit{x}}^{\prime }\mid$, $\overline{r}=\mid \mathit{x}-{\overline{\mathit{x}}}^{\prime }\mid$, ${\overline{\mathit{x}}}^{\prime }=\left(x^{\prime },-y^{\prime }\right)$ and ${\mathrm{H}}_0^{(1)}$ is the 0-th Hankel function of the first kind.

Green’s function (7) satisfies the following equations

$$\begin{array}{ccc}& & \Delta G_N+k_0^2{\mu }_r{G}_N=-\delta (\mathbf{x},{\mathbf{x}}^{\prime }),\mathbf{x},{\mathbf{x}}^{\prime }\in {\mathbb{R}}_{+}^2,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & \frac{\partial G_N}{\partial n}=0,y^{\prime }=0\hfill \end{array}$$

(9)

and the Sommerfeld radiation condition

$$\underset{r\to \infty }{lim}\sqrt{r}\left(\frac{\partial G_N}{\partial r}-\mathrm{i}{k}_0{G}_N\right)=0,$$

where $\delta$ is the Dirac delta function.

Multiplying $G_N$ on Equation (4) and $u^s$ on Equation (8), we can derive an equation for $G_N$ and $u^s$

$${\int \int }_{{\mathsf{\Omega }}_r^{+}}\left(G_N\Delta u^s-u^s\Delta G_N\right)d\mathsf{\Omega }={\int \int }_{{\mathsf{\Omega }}_r^{+}}{u}^s\delta (\mathit{x},{\mathit{x}}^{\prime })d\mathsf{\Omega }.$$

(10)

According to the property of $\delta$—function and Green’s formulation, we have

$$u^s(\mathbf{x})={\int }_{\mathsf{\Gamma }}\left(G_N\left({\mathbf{x}}^{\prime },\mathbf{x}\right)\frac{\partial u^s\left({\mathbf{x}}^{\prime }\right)}{\partial n\left({\mathbf{x}}^{\prime }\right)}-u^s\left({\mathbf{x}}^{\prime }\right)\frac{\partial G_N\left({\mathbf{x}}^{\prime },\mathbf{x}\right)}{\partial n\left({\mathbf{x}}^{\prime }\right)}\right)ds({\mathbf{x}}^{\prime }).$$

(11)

A detailed derivation is available in [3].

In terms of Equations (5) and (9), the above equation can be written as

$$u^s(\mathit{x})={\int }_{\mathsf{\Gamma }}{\left[G_N\left({\mathit{x}}^{\prime },\mathit{x}\right)\frac{\partial u^s\left({\mathit{x}}^{\prime }\right)}{\partial n\left({\mathit{x}}^{\prime }\right)}\right]}_{y^{\prime }=0^{+}}ds({\mathit{x}}^{\prime })=-{\int }_{\mathsf{\Gamma }}{\left[G_N\left({\mathit{x}}^{\prime },\mathit{x}\right)\frac{\partial u^s\left({\mathit{x}}^{\prime }\right)}{\partial y^{\prime }}\right]}_{y^{\prime }=0^{+}}d{x}^{\prime }.$$

(12)

Substituting Equations (1) and (7) into the above equation, we obtain

$$u(x)=2e^{\mathrm{i}\alpha x}-\frac{\mathrm{i}}{2}{\int }_{\mathsf{\Gamma }}{H}_0^{(1)}\left(k_0\mid x-x^{\prime }\mid \right)\frac{\partial u\left(x^{\prime }\right)}{\partial y^{\prime }}d{x}^{\prime },$$

where $H_0^{(1)}(v)=J_0(v)+\mathrm{i}{Y}_0(v)$. $J_0(v),Y_0(v)$ are Bessel functions of the first and second kinds of order 0 respectively.

Noting the continuity conditions on the aperture $\mathsf{\Gamma }$

$$u(x,y){\mid }_{y=0^{+}}=u(x,y){\mid }_{y=0^{-}},\frac{\partial u}{\partial n}{\mid }_{y=0^{+}}=\frac{1}{{\epsilon }_r}\frac{\partial u}{\partial n}{\mid }_{y=0^{-}},$$

We can obtain the transparent boundary on $\mathsf{\Gamma }$

$$u(x)=2e^{\mathrm{i}\alpha x}-\frac{\mathrm{i}}{2}{\int }_{\mathsf{\Gamma }}{H}_0^{(1)}\left(k_0\mid x-x^{\prime }\mid \right)\frac{1}{{\epsilon }_r}\frac{\partial u\left({\mathit{x}}^{\prime }\right)}{\partial y^{\prime }}{\mid }_{y^{\prime }=0^{-}}d{x}^{\prime }.$$

Let $g(x)=2e^{\mathrm{i}\alpha x}$, the transparent boundary condition on $\mathsf{\Gamma }$ in TE polarization can be simplified as

$$u=\mathbb{T}\left(\frac{1}{{\epsilon }_r}\frac{\partial u}{\partial y}\right)+g(x),$$

(13)

where $\mathbb{T}\left(\frac{1}{{\epsilon }_r}\frac{\partial u}{\partial y}\right)=-\frac{\mathrm{i}}{2}{\int }_{\mathsf{\Gamma }}{H}_0^{(1)}\left(k_0\mid x-x^{\prime }\mid \right)\frac{1}{{\epsilon }_r}\frac{\partial u\left({\mathit{x}}^{\prime }\right)}{\partial y^{\prime }}{\mid }_{y^{\prime }=0^{-}}d{x}^{\prime }$.

Hence, the electromagnetic scattering from the single cavity in TE polarization can be transformed into the following equations in a bounded domain [28].

$$\begin{array}{cccc}\hfill & \nabla ·\left(\frac{1}{{\epsilon }_r}\nabla u\right)+k_0^2{\mu }_{r}u=0,\hfill & \hfill \mathrm{in}& \mathsf{\Omega },\hfill \\ \hfill & \frac{1}{{\epsilon }_r}\frac{\partial u}{\partial n}=0,\hfill & \hfill \mathrm{on}& (\partial \mathsf{\Omega }\backslash \mathsf{\Gamma })\cup {\mathsf{\Gamma }}^c,\hfill \\ \hfill & u=\mathbb{T}\left(\frac{1}{{\epsilon }_r}\frac{\partial u}{\partial n}\right)+g(x),\hfill & \hfill \mathrm{on}& \mathsf{\Gamma }.\hfill \end{array}$$

(14)

The existence and uniqueness of solutions of (14) for arbitrary wave numbers have been obtained in [26].

3. Fast High-Order Schemes for the Scattering from Single Cavity

3.1. Second-Order Finite Difference Scheme

Let ${\displaystyle {\{x_i,y_j\}}_{i,j=0}^{M+1,N+1}}$ define a uniform partition on $\mathsf{\Omega }=[0,a]\times [-b,0]$ with $x_i=i{h}_x,i=0,1,\dots ,M+1$, $y_j=-b+j{h}_y,j=0,1,\dots ,N+1$, $h_x=a/(M+1)$, $h_y=b/(N+1)$. For simplicity, we consider the case of $h_x=h_y=h$. It can be generalized to the case of $h_x\ne h_y$. Denote $u_{i,j}$ as the numerical solution at point $(x_i,y_j)$. The second-order finite difference approximation inside the cavity can be written as

$$\frac{1}{{\epsilon }_r}\left({\delta }_x^2+{\delta }_y^2\right)u_{i,j}+k_0^2{\mu }_r{u}_{i,j}=f_{i,j},i=0,1,\dots ,M+1,j=0,1,\dots ,N+1,$$

(15)

where ${\delta }_x^2,{\delta }_y^2$ are second-order finite difference operators.

Considering the second-order approximation for Neumann boundary condition on $y=-b$

$$\frac{u_{i,1}-u_{i,-1}}{2h}=\frac{\partial u}{\partial n}{\mid }_{y=-b}+O(h^2).$$

Since $\frac{\partial u}{\partial n}=0$, we have $u_{i,-1}=u_{i,1},i=0,1,\dots ,M+1$. The Neumann boundary conditions on the left and right boundaries can be handled analogously.

Denote all the unknowns by:

$$\begin{array}{cc}\hfill & U=(u_{0,0},u_{0,1},\dots ,u_{0,N},u_{1,0},u_{1,1},\dots ,u_{1,N},\dots ,u_{M+1,0},u_{M+1,1},\dots ,u_{M+1,N}),\hfill \\ & u_{N+1}=(u_{0,N+1},u_{1,N+1},\dots ,u_{M+1,N+1}).\hfill \end{array}$$

Therefore, we can rewrite the second-order scheme (15) as the matrix form

$$\left(\frac{1}{{\epsilon }_r}\left(A_x\otimes I_{N+1}+I_{M+2}\otimes A_y\right)+k_0^2{\mu }_{r}I\right)U+\frac{1}{{\epsilon }_r}(I_{M+2}\otimes a_{N+1})u_{:,N+1}=F,$$

(16)

where

$$A_x=\frac{1}{h^2}\left(\begin{array}{ccccc}-2& 2& & & \\ 1& -2& 1& & \\ & \ddots & \ddots & \ddots & \\ & & 1& -2& 1\\ & & & 2& -2\end{array}\right),A_y=\frac{1}{h^2}\left(\begin{array}{ccccc}-2& 2& & & \\ 1& -2& 1& & \\ & \ddots & \ddots & \ddots & \\ & & 1& -2& 1\\ & & & 1& -2\end{array}\right),$$

$$\begin{array}{c}{a}_{N+1}=\frac{1}{h^2}{(0,\dots ,0,1)}^T,\hfill \\ F={\left(f_{0,0},f_{0,1},\dots ,f_{0,N},f_{1,0},f_{1,1},\dots ,f_{1,N},\dots ,f_{M+1,0},f_{M+1,1},\dots ,f_{M+1,N}\right)}^T.\hfill \end{array}$$

Here ⊗ is the Kronecker product. $I_{M+2}$ and $I_{N+1}$ are identity matrix in $(M+2)\times (M+2)$ dimension and $(N+1)\times (N+1)$ dimensions respectively, I is identity matrix in $(M+2)(N+1)$×$(M+2)(N+1)$ dimensions.

Below we consider the approximations of the transparent boundary in (13). The nonlocal integral operator in (13) can be rewritten by

$$\begin{array}{cc}\hfill \mathbb{T}\left(\frac{\partial u}{\partial y}\right)=& \frac{1}{2}{\int }_{\mathsf{\Gamma }}{Y}_0\left(k_0\mid x-x^{\prime }\mid \right)\frac{\partial u\left({\mathit{x}}^{\prime }\right)}{\partial y^{\prime }}{\mid }_{y^{\prime }=0^{-}}d{x}^{\prime }\hfill \\ \hfill & -\frac{\mathrm{i}}{2}{\int }_{\mathsf{\Gamma }}{J}_0\left(k_0\mid x-x^{\prime }\mid \right)\frac{\partial u\left({\mathit{x}}^{\prime }\right)}{\partial y^{\prime }}{\mid }_{y^{\prime }=0^{-}}d{x}^{\prime }.\hfill \end{array}$$

(17)

Since ${\displaystyle Y_0(v)\sim \frac{2}{\pi }ln\frac{v}{2}\mathrm{as}v\to 0}$, the first part of Equation (17) contains the weakly singular integral.

We adopt a piecewise linear approximation for the first item and a trapezoidal formulation for the second item of Equation (17). So the nonlocal integral operator $\mathbb{T}(z)$ can be approximated as follows

$$\mathbb{T}(z)\approx \sum _{j=1}^M{T}_{ij}{z}_j,$$

where

$$\begin{array}{cc}\hfill & T_{ij}=T_{ij}^{\mathrm{re}}+\mathrm{i}{T}_{ij}^{\mathrm{im}},\hfill \\ \hfill & T_{ij}^{\mathrm{re}}=\frac{1}{2}{\int }_{\mathsf{\Gamma }}{Y}_0\left(k_0\mid x_i-x_j\mid \right){\varphi }_j^1\left(x^{\prime }\right)d{x}^{\prime },\hfill \\ \hfill & T_{ij}^{\mathrm{im}}=-\frac{h}{2}{J}_0\left(k_0\mid x_i-x_j\mid \right),\hfill \end{array}$$

and ${\varphi }_j^1\left(x^{\prime }\right)$ are the piecewise linear basis functions.

The second-order approximation for (13) is given as

$$u_{:,N+1}=T\frac{1}{{\epsilon }_r}\frac{u_{:,N+2}-u_{:,N}}{2h}+g(x_{:,N+1}).$$

(18)

Moreover, $u_{:,N+1}$ satisfies the second-order scheme (15)

$$\frac{1}{{\epsilon }_r}\left({\delta }_x^2+{\delta }_y^2\right)u_{i,N+1}+k_0^2{u}_{i,N+1}=f_{i,N+1},i=0,1,\dots ,M+1.$$

(19)

Rewrite (19) into matrix form [10].

$$\left(\frac{1}{{\epsilon }_r}\left(A_x-\frac{2}{h^2}{I}_{M+2}\right)+k_0^2\right)u_{:,N+1}+\frac{1}{{\epsilon }_r{h}^2}{u}_{:,N}+\frac{1}{{\epsilon }_r{h}^2}{u}_{:,N+2}=F_{:,N+1}.$$

(20)

Combining (18) and (20), we have

$$2T{u}_{:,N}+(2h{\epsilon }_r{I}_{M+2}+T{J}_2^{-1}{J}_1)u_{:,N+1}=T{J}_2^{-1}{F}_{:,N+1}+2h{\epsilon }_{r}g,$$

(21)

where $J_1=\frac{1}{{\epsilon }_r}\left(A_x-\frac{2}{h^2}{I}_{M+2}\right)+k_0^2{I}_{M+2}$, $J_2=\frac{1}{{\epsilon }_r{h}^2}{I}_{M+2}$.

Then we derive the reduced linear system

$$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\left(\begin{array}{c}U\\ u_{:,N+1}\end{array}\right)=\left(\begin{array}{c}F\\ F_{\mathsf{\Gamma }}\end{array}\right),$$

where

$$\begin{array}{cc}\hfill & A_{11}=\frac{1}{{\epsilon }_r}\left(A_x\otimes I_{N+1}+I_{M+2}\otimes A_y\right)+k_0^2{\mu }_{r}I,\hfill \\ \hfill & A_{12}=\frac{1}{{\epsilon }_r}(I_{M+2}\otimes a_{N+1}),\hfill \\ \hfill & A_{21}=2T\otimes a_{N+1}^T,A_{22}=2h{\epsilon }_r{I}_{M+2}+T{J}_2^{-1}{J}_1,\hfill \\ \hfill & F_{\mathsf{\Gamma }}=T{J}_2^{-1}{F}_{:,N+1}+2h{\epsilon }_{r}g.\hfill \end{array}$$

3.2. The Fast Second-Order Algorithm for the Scattering from the Single Cavity

For the matrix $A_x$, we have

$$C_x^{-1}{A}_x{C}_x=\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{M+2}),$$

where $C_x$ is the discrete cosine transformation matrix,

$${(C_x)}_{ij}=cos\frac{\pi ij}{M+1},{\lambda }_i=2cos\left(\frac{i\pi }{M+1}\right)-2,i,j=0,1,2,\dots ,M+1.$$

Multiplying $C_x^{-1}\otimes I_{N+1}$ on both sides of Equation (16), we have

$$\frac{1}{{\epsilon }_r}\left(\mathsf{\Lambda }\otimes I_{N+1}+I_{M+2}\otimes A_y\right)\overline{U}+k_0^2{\mu }_r\overline{U}+\frac{1}{{\epsilon }_r}(I_{M+2}\otimes a_{N+1}){\overline{u}}_{:,N+1}=\overline{F},$$

(22)

where

$$\begin{array}{cc}\hfill & \overline{U}=(C_x^{-1}\otimes I_{N+1})u\hfill \\ \hfill =& {({\overline{u}}_{0,0},{\overline{u}}_{0,1},\cdots ,{\overline{u}}_{0,N},{\overline{u}}_{1,0},{\overline{u}}_{1,1},\cdots ,{\overline{u}}_{1,N},\cdots ,{\overline{u}}_{M+1,0},{\overline{u}}_{M+1,1},\cdots ,{\overline{u}}_{M+1,N})}^T,\hfill \\ \hfill & \overline{F}=(C_x^{-1}\otimes I_{N+1})F\hfill \\ \hfill =& {\left({\overline{f}}_{0,0},{\overline{f}}_{0,1},\cdots ,{\overline{f}}_{0,N},{\overline{f}}_{10},{\overline{f}}_{11},\cdots ,{\overline{f}}_{1N},\cdots ,{\overline{f}}_{M+1,0},{\overline{f}}_{M+1,1},\cdots ,{\overline{f}}_{M+1,N}\right)}^T.\hfill \\ \hfill & {\overline{u}}_{:,N+1}=C_x^{-1}{u}_{:,N+1}={\left({\overline{u}}_{0,N+1},{\overline{u}}_{1,N+1},\cdots ,{\overline{u}}_{M+1,N+1}\right)}^T,\hfill \end{array}$$

Moreover, Equation (22) can be written as $M+2$ systems

$$\left(\frac{1}{{\epsilon }_r}({\lambda }_i{I}_{N+1}+A_y)+k_0^2{\mu }_r{I}_{N+1}\right){\overline{U}}_{i,:}+\frac{1}{{\epsilon }_r}{a}_{N+1}{\overline{u}}_{i,N+1}={\overline{F}}_{i,:},i=0,1,\cdots ,M+1,$$

(23)

where

$${\overline{U}}_{i,:}={({\overline{u}}_{i,0},{\overline{u}}_{i,1},\dots ,{\overline{u}}_{i,N})}^T,{\overline{F}}_{i,:}={({\overline{f}}_{i,0},{\overline{f}}_{i,1},\dots ,{\overline{f}}_{i,N})}^T.$$

As can be seen from Equation (23), $\frac{1}{{\epsilon }_r}({\lambda }_i{I}_{N+1}+A_y)+k_0^2{\mu }_r{I}_{N+1}$ is a tridiagonal matrix. For small enough h, we can derive the $LU$ decomposition

$$\frac{1}{{\epsilon }_r}({\lambda }_i{I}_{N+1}+A_y)+k_0^2{\mu }_r{I}_{N+1}={\mathbb{L}}_i{\mathbb{U}}_i,i=0,1,2,\dots ,M+1.$$

Multiplying ${\mathbb{L}}_i^{-1}$ on both sides of Equation (23), we have

$${\mathbb{U}}_i{\overline{U}}_{i,:}+\frac{1}{{\epsilon }_r}{\mathbb{L}}_i^{-1}{a}_{N+1}{\overline{u}}_{i,N+1}={\mathbb{L}}_i^{-1}{\overline{F}}_{i,:}.$$

Extracting the last elements from the above systems, we can obtain

$$p_i{\overline{u}}_{i,N}+q_i{\overline{U}}_{i,N+1}={\tilde{f}}_{i,N},i=0,1,\dots ,M+1,$$

where $p_i$ and $q_i$ are the last elements of ${\mathbb{U}}_i$ and ${\mathbb{L}}_i^{-1}{a}_{N+1}/{\epsilon }_r$ respectively, and ${\tilde{f}}_{i,N}$ is the last element of ${\mathbb{L}}_i^{-1}{\overline{F}}_{i,:}$.

For $i=0,1,\dots ,M+1$, we have

$$P{\overline{u}}_{:,N}+Q{\overline{u}}_{:,N+1}=\tilde{F},$$

where $P=\mathrm{diag}(p_1,p_2,\dots ,p_{M+2})$, $Q=\mathrm{diag}(q_1,q_2,\dots ,q_{M+2})$ and $\tilde{F}=({\tilde{f}}_{0,N},{\tilde{f}}_{1,N},\dots ,$${\tilde{f}}_{M+1,N}{)}^T$.

Combining with Equation (21), we can derive the reduced formulation only with the unknowns on the aperture

$$\left(2h{\epsilon }_r{C}_x+T{C}_x{J}_2^{-1}{J}_1-2T{C}_x{P}^{-1}Q\right){\overline{u}}_{:,N+1}=2h{\epsilon }_{r}g+T{C}_x{J}_2^{-1}{\overline{F}}_{:,N+1}-2T{C}_x{P}^{-1}\tilde{F}.$$

(24)

We analyze the computational cost of Algorithm 1. The matrix T is calculated by using the traditional trapezoidal formula, and the cost of Step (i) is $O(M+2)$. In Step (ii), we need to calculate the $LU$-decomposition for $({\lambda }_i{I}_{N+1}+A_y)/{\epsilon }_r+k_0^2{I}_{N+1}$. Noting the tridiagonal structure of these matrices, we only need $O((M+2)(N+1))$ operations if $f\ne 0$. The cost of calculating the product of the matrix $C_x$ and a complex vector is $O({(M+2)}^2(N+1))$ in Step (iii). In Step (iv), we use the BiCG method for solving the linear systems (24). Each iteration only needs to calculate one matrix-vector multiplication, and the cost of each iteration is $O(24p(M+2)log(M+2))$, p is the number of iterations.

Algorithm 1: The procedure of the second-order fast algorithm for scattering from the single cavity (The Fast Algorithm for the Scattering Problem From the Single Cavity).

Step (i) Generate the matrix T. Step (ii) Calculate the $LU$ decomposition to get P and Q by using the forward Gaussian elimination with a row partial pivoting. Step (iii) Calculate $C_{x}F$. Step (iv) Solve the system (24) for ${\overline{u}}_{:,N+1}$. Step (v) Solve the system (22) for rest of the unknowns.

3.3. Fourth-Order Finite Difference Scheme

The fourth-order compact finite difference scheme can be written as

$$\begin{array}{cc}\hfill \left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left({\delta }_x^2+{\delta }_y^2\right)u_{i,j}+& \frac{h^2}{6}\frac{1}{{\epsilon }_r}{\delta }_x^2{\delta }_y^2{u}_{i,j}+k_0^2{u}_{i,j}=f_{i,j}+\frac{h^2}{12}\Delta f_{i,j},\hfill \\ \hfill i=& 0,1,\dots ,M+1,j=0,1,\dots ,N+1.\hfill \end{array}$$

(25)

$E_{i,j}$ is the leading truncation error and is given by

$$\begin{array}{cc}\hfill E_{i,j}=& \left(\frac{k^2}{144}{\left(u_{x^4}+u_{y^4}\right)}_{i,j}+\frac{1}{360{\epsilon }_r}{\left(u_{x^6}+u_{y^6}\right)}_{i,j}+\frac{1}{72{\epsilon }_r}{\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)}_{i,j}\right)h^4\hfill \\ \hfill & -\left(\frac{1}{144}{\left(f_{x^4}+f_{y^4}\right)}_{i,j}\right)h^4.\hfill \end{array}$$

Otherwise, Equation (25) can be written as

$$\begin{array}{cc}\hfill c_2{u}_{i-1,j-1}& +c_1{u}_{i-1,j}+c_2{u}_{i-1,j+1}+c_1{u}_{i,j-1}+c_0{u}_{i,j}+c_1{u}_{i,j+1}\hfill \\ \hfill & +c_2{u}_{i+1,j-1}+c_1{u}_{i+1,j}+c_2{u}_{i+1,j+1}=F_{i,j},\hfill \end{array}$$

(26)

where $c_0=\frac{2k^2{h}^2}{3}-\frac{10}{3{\epsilon }_r}$, $c_1=\frac{k^2{h}^2}{12}+\frac{2}{3{\epsilon }_r}$, $c_2=\frac{1}{6{\epsilon }_r}$, $F_{i,j}=h^2{f}_{i,j}+\frac{h^4}{12}\Delta f_{i,j}$.

Next, we consider the Neumann boundary condition on the boundary S. By Taylor’s expansion, we can obtain the fourth-order approximation for $\frac{\partial u}{\partial n}=0$ on $y=-b$

$$\frac{\partial u}{\partial y}{\mid }_{y=-b}=\frac{u_{i,1}-u_{i,-1}}{2h}-\frac{h^2}{6}{\left(u_{yyy}\right)}_{i,0}+O\left(h^4\right).$$

Since $\frac{\partial u}{\partial y}{\mid }_{y=-b}=0$ and $u_{yyy}={\epsilon }_r{f}_y-k^2{\epsilon }_r{u}_y-u_{xxy}$, we have

$$\left(\frac{1}{{\epsilon }_r}+\frac{h^2{k}^2}{6}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,1}-\left(\frac{h^2{k}^2}{6{\epsilon }_r}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,-1}=\frac{h^3}{3}{(f_y)}_{i,0},i=0,1,\dots ,M+1.$$

(27)

Moreover, Equation (25) can be written as

$$\begin{array}{cc}\hfill & \left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,j+1}+\left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(h^2{\delta }_x^2-2\right)-\frac{h^2}{3{\epsilon }_r}{\delta }_x^2+k^2{h}^2\right)u_{i,j}\hfill \\ \hfill +& \left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,j-1}\hfill \\ \hfill =& h^2{f}_{i,j}+\frac{h^4}{12}\Delta f_{i,j},i=0,1,\dots ,M+1,j=0,1,\dots ,N+1.\hfill \end{array}$$

Let $j=0$, we have

$$\begin{array}{cc}\hfill & \left(\frac{1}{{\epsilon }_r}+\frac{k_0^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,1}+\left(\left(\frac{1}{{\epsilon }_r}+\frac{k_0^2{h}^2}{12}\right)\left(h^2{\delta }_x^2-2\right)-\frac{h^2}{3{\epsilon }_r}{\delta }_x^2+k_0^2{h}^2\right)u_{i,0}\hfill \\ \hfill +& \left(\frac{1}{{\epsilon }_r}+\frac{k_0^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,-1}\hfill \\ \hfill =& h^2{f}_{i,j}+\frac{h^4}{12}\Delta f_{i,j},i=0,1,\dots ,M+1.\hfill \end{array}$$

(28)

Combining Equations (27) and (28), we have

$$\begin{array}{cc}\hfill & 2\left(\frac{1}{{\epsilon }_r}+\frac{k_0^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\delta }_x^2\right)u_{i,1}+\left(\left(\frac{1}{{\epsilon }_r}+\frac{k_0^2{h}^2}{12}\right)\left(h^2{\delta }_x^2-2\right)-\frac{h^2}{3{\epsilon }_r}{\delta }_x^2+k_0^2{h}^2\right)u_{i,0}\hfill \\ \hfill =& h^2{f}_{i,0}+\frac{h^3}{3}{(f_y)}_{i,0}+\frac{h^4}{12}\Delta f_{i,0},i=0,1,\dots ,M+1.\hfill \end{array}$$

Similarly, we can derive the fourth-order approximation of the Neumann boundary condition on the other side of the boundaries. Therefore, the compact fourth-order scheme can be written as the following matrix form

$$\begin{array}{cc}\hfill & \left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(A_x\otimes I_{N+1}+I_{M+2}\otimes A_y\right)+\frac{h^2}{6{\epsilon }_r}\left(A_x\otimes A_y\right)+k^{2}I\right)U_1\hfill \\ \hfill & +\left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(I_{M+2}\otimes a_{N+1}\right)+\frac{h^2}{6{\epsilon }_r}\left(A_x\otimes a_{N+1}\right)\right)u_{:,N+1}\hfill \\ \hfill =& F_1+\frac{h^2}{12}\Delta F_1+\frac{1}{12}\left(I_{M+2}\otimes b_{N+1}\right)f_{:,0}+\frac{1}{12}\left(b_{N+1}\otimes I_{M+2}\right)f_{0,:}\hfill \\ \hfill & +\frac{1}{12}\left(a_{N+1}\otimes I_{M+2}\right)f_{M+1,:},\hfill \end{array}$$

(29)

where

$$F_1={\left(f_{0,0},f_{0,1},\cdots ,f_{0,N},f_{1,0},f_{1,1},\cdots ,f_{1,N},\cdots ,f_{M+1,0},f_{M+1,1},\cdots ,f_{M+1,N}\right)}^T,$$

$$U_1=(u_{0,0},u_{0,1},\cdots ,u_{0,N},u_{1,0},u_{1,1},\cdots ,u_{1,N},\cdots ,u_{M+1,0},u_{M+1,1},\cdots ,u_{M+1,N}),$$

$$b_{N+1}=\frac{1}{h^2}{(1,0,\cdots ,0)}^T.$$

Below we need to deal with the transparent boundary condition on the aperture in particular. Firstly, By means of Taylor’s expansion, we have

$$\frac{\partial u}{\partial y}=\frac{u_{i,N+2}-u_{i,N}}{2h}-\frac{h^2}{6}\left({\epsilon }_r{f}_y-k^2{\epsilon }_r{u}_y-u_{xxy}\right){\mid }_{i,N+1}+O(h^4),i=0,1,\dots ,M+1.$$

Then in order to eliminate the ghost points, we introduce the variable $\nu$

$$\begin{array}{cc}\hfill u_y=& \frac{u_{i,N+2}-u_{i,N}}{2h}+\nu h^2{u}_{xxy}+O(h^2)\hfill \\ \hfill =& \frac{(1-2\nu )(u_{i,N+2}-u_{i,N})}{2h}+\frac{\nu (u_{i+1,N+2}+u_{i-1,N+2}-u_{i+1,N}-u_{i-1,N})}{2h}.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{cc}\hfill \frac{\partial u}{\partial y}=& \frac{u_{i,N+2}-u_{i,N}}{2h}-\frac{h^2}{6}{\left({\epsilon }_r{f}_y\right)}_{i,N+1}\hfill \\ \hfill & +\frac{1}{6}\frac{(u_{i+1,N+2}-2u_{i,N+2}+u_{i-1,N+2})-(u_{i+1,N}-2u_{i,N}+u_{i-1,N})}{2h}\hfill \\ \hfill & +\frac{{\epsilon }_r{k}^2{h}^2}{6}\left(\frac{(1-2\nu )(u_{i,N+2}-u_{i,N})}{2h}+\frac{\nu (u_{i+1,N+2}+u_{i-1,N+2}-u_{i+1,N}-u_{i-1,N})}{2h}\right).\hfill \end{array}$$

Combing with the transparent boundary condition (13), we have

$$\begin{array}{cc}\hfill \sum _{l=1}^{M+2}{\vartheta }_{i,l}{u}_{l,N+1}-{\widehat{g}}_i+\frac{h^3}{3}{\left(f_y\right)}_{i,N+1}=& \left(\frac{2}{3{\epsilon }_r}+\frac{k^2{h}^2}{6}(1-2\nu )\right)(u_{i,N+2}-u_{i,N})\hfill \\ \hfill & +\left(\frac{1}{6{\epsilon }_r}+\nu \frac{k^2{h}^2}{6}\right)(u_{i+1,N+2}+u_{i-1,N+2})\hfill \\ \hfill & -\left(\frac{1}{6{\epsilon }_r}+\nu \frac{k^2{h}^2}{6}\right)(u_{i+1,N}+u_{i-1,N}),\hfill \end{array}$$

(30)

where ${\vartheta }_{i,l}=2h{T}_{i,l}^{-1},{\widehat{g}}_i=2h{\sum }_{l=1}^{M+2}{T}_{i,l}^{-1}g$.

Take $j=N+1$ for the fourth-order scheme (25), we have

$$\begin{array}{cc}\hfill & c_1(u_{i,N+2}+u_{i,N})+c_2(u_{i+1,N+2}+u_{i-1,N+2}+u_{i+1,N}+u_{i-1,N})\hfill \\ \hfill & +c_1(u_{i+1,N+1}+u_{i-1,N-1})+c_0{u}_{i,N+1}\hfill \\ \hfill =& h^2{f}_{:,N+1}+\frac{h^4}{12}\Delta f_{:,N+1}.\hfill \end{array}$$

(31)

Moreover, Equation (30) can be written as

$$\begin{array}{cc}\hfill & {\tilde{c}}_1(u_{i,N+2}-u_{i,N})+{\tilde{c}}_2(u_{i+1,N+2}+u_{i-1,N+2}-u_{i+1,N}-u_{i-1,N})-\sum _{l=1}^{M+2}{\vartheta }_{i,l}{u}_{l,N+1}\hfill \\ \hfill =& \frac{h^3}{3}{\left(f_y\right)}_{i,N+1}-{\widehat{g}}_i,\hfill \end{array}$$

(32)

where ${\tilde{c}}_1=\frac{2}{3{\epsilon }_r}+\frac{k^2{h}^2}{6}(1-2\nu )$, ${\tilde{c}}_2=\frac{1}{6{\epsilon }_r}+\nu \frac{k^2{h}^2}{6}$.

To eliminate $u_{i,N+2},u_{i-1,N+2},u_{i+1,N+2}$, we define a constant $\eta$ such that

$$\eta =\frac{c_1}{{\tilde{c}}_1}=\frac{c_2}{{\tilde{c}}_2}.$$

We have computed $\nu k^2{h}^2=\frac{\left(\frac{4}{{\epsilon }_r}+k^2{h}^2\right)c_2-\frac{1}{{\epsilon }_r}{c}_1}{c_1+2c_2}$, hence

$$\begin{array}{cc}\hfill & {\tilde{c}}_1=\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{6}\right)\frac{c_1}{c_1+2c_2},\hfill \\ \hfill & {\tilde{c}}_2=\frac{c_2}{c_1}{\tilde{c}}_1,\hfill \\ \hfill & \eta ={\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{6}\right)}^{-1}(c_1+2c_2).\hfill \end{array}$$

Therefore, combing Equations (26) and (32), the reduced equation can be written as

$$\begin{array}{cc}\hfill & \eta \sum _{l=1}^{M+2}{\vartheta }_{i,l}{u}_{l,N+1}+2c_1{u}_{i,N}+2c_2(u_{i+1,N}+u_{i-1,N})\hfill \\ \hfill & +c_1(u_{i+1,N+1}+u_{i-1,N+1})+c_0{u}_{i,N+1}\hfill \\ \hfill =& h^2{f}_{:,N+1}+\frac{h^4}{12}\Delta f_{:,N+1}+\eta {\widehat{g}}_i-\eta \frac{h^3}{3}{\left(f_y\right)}_{i,N+1}.\hfill \end{array}$$

Denote

$$\begin{array}{cc}\hfill & L_1=\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)I_{M+2}+\frac{h^2}{6{\epsilon }_r}{A}_x,\hfill \\ \hfill & L_2=\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(h^2{A}_x-2I_{M+2}\right)-\frac{h^2}{3{\epsilon }_r}{A}_x+k^2{h}^2{I}_{M+2}.\hfill \end{array}$$

The above equation can be written in matrix form

$$\begin{array}{cc}\hfill & 2T{L}_1{u}_{:,N}+\left(2h\eta I_{M+2}+T{L}_2\right)u_{:,N+1}\hfill \\ \hfill =& 2h\eta g+T\left(h^2{F}_{:,N+1}+\frac{h^4}{12}\Delta F_{:,N+1}-\eta \frac{h^3}{3}{(f_y)}_{:,N+1}\right)\hfill \end{array}$$

(33)

Then we derive the reduced linear system

$$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\left(\begin{array}{c}U\\ u_{:,N+1}\end{array}\right)=\left(\begin{array}{c}F\\ F_{\mathsf{\Gamma }}\end{array}\right),$$

(34)

where

$$\begin{array}{cc}\hfill A_{11}=& \left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(A_x\otimes I_N+I_M\otimes A_y\right)+\frac{h^2}{6}\frac{1}{{\epsilon }_r}\left(A_x\otimes A_y\right)+k^{2}I,\hfill \\ \hfill A_{12}=& \left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(I_M\otimes a_{N+1}\right)+\frac{h^2}{6}\frac{1}{{\epsilon }_r}\left(A_x\otimes a_{N+1}\right),\hfill \\ \hfill A_{21}=& 2T{L}_1\otimes a_{N+1}^T,A_{22}=2h{I}_{M+2}+T{L}_2,\hfill \\ \hfill F=& F_1+\frac{h^2}{12}\Delta F+\frac{h}{3}\left({\tilde{I}}_{M+2}\otimes b_{N+1}\right){(f_y)}_{:,0}+\frac{h}{3}\left(b_{N+1}\otimes I_{M+2}\right){(f_x)}_{0,:}\hfill \\ \hfill & +\frac{h}{3}\left(a_{N+1}\otimes I_{M+2}\right){(f_x)}_{M+1,:},\hfill \\ \hfill F_{\mathsf{\Gamma }}=& 2h\eta g+T\left(h^2{F}_{:,N+1}+\frac{h^4}{12}\Delta F_{:,N+1}-\eta \frac{h^3}{3}{(f_y)}_{:,N+1}\right),\hfill \\ \hfill {\tilde{I}}_{M+2}=& \mathrm{diag}(0,1,\dots ,1,0).\hfill \end{array}$$

3.4. The Fast Fourth-Order Algorithm for the Scattering from the Single Cavity

Multiple $C_x^{-1}\otimes I_{N+1}$ on both side of (29), we have

$$\begin{array}{cc}\hfill & \left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(\mathsf{\Lambda }\otimes I_{N+1}+I_{M+2}\otimes A_y\right)+\frac{h^2}{6{\epsilon }_r}\left(\mathsf{\Lambda }\otimes A_y\right)+k^{2}I\right){\overline{U}}_1\hfill \\ \hfill +& \left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left(I_{M+2}\otimes a_{N+1}\right)+\frac{h^2}{6{\epsilon }_r}\left(A_x\otimes a_{N+1}\right)\right){\overline{u}}_{:,N+1}=\overline{F},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \overline{F}=& \left(C_x^{-1}\otimes I_{N+1}\right)\left(F_1+\frac{h^2}{12}\Delta F_1+\frac{1}{12}\left(I_{M+2}\otimes b_{N+1}\right)f_{:,0}+\frac{1}{12}\left(b_{N+1}\otimes I_{M+2}\right)f_{0,:}\right)\hfill \\ \hfill & +\left(C_x^{-1}\otimes I_{N+1}\right)\left(\frac{1}{12}\left(a_{N+1}\otimes I_{M+2}\right)f_{M+1,:}\right).\hfill \end{array}$$

Therefore, we have $M+2$ linear systems which are written as

$$\begin{array}{cc}\hfill & \left(\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left({\lambda }_i{I}_{N+1}+A_y\right)+\frac{h^2}{6{\epsilon }_r}\left({\lambda }_i{A}_y\right)+k_0^2{I}_{N+1}\right){\overline{u}}_{i,:}\hfill \\ \hfill & +\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\lambda }_i\right)a_{N+1}{\overline{u}}_{i,N+1}={\overline{F}}_{i,:}.\hfill \end{array}$$

Let $\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}\right)\left({\lambda }_i{I}_{N+1}+A_y\right)+\frac{h^2}{6{\epsilon }_r}{\lambda }_i{A}_y+k_0^2{I}_{N+1}={\mathbb{L}}_i{\mathbb{U}}_i$ be the LU-decomposition, for enough small h, we have

$${\mathbb{U}}_i{\overline{u}}_{i,:}+{\mathbb{L}}_i^{-1}\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}+\frac{h^2}{6{\epsilon }_r}{\lambda }_i\right)a_{N+1}{\overline{u}}_{i,N+1}={\mathbb{L}}_i^{-1}{\overline{F}}_{i,:}.$$

Take the last element of the above equation, we have the following relationship between $u_N$ and $u_{N+1}$

$${\overline{u}}_{:,N}=-P^{-1}Q{\overline{u}}_{:,N+1}+P^{-1}{\mathbb{L}}_i^{-1}{\overline{F}}_{:,N},$$

(35)

where $P=\mathrm{diag}(p_1,p_2,\dots ,p_{M+2}),Q=\mathrm{diag}(q_1,q_2,\dots ,q_{M+2})$, $p_i$ is the last entry of ${\mathbb{U}}_i$, and $q_i$ is the last element of ${\mathbb{L}}_i^{-1}\left(\frac{1}{{\epsilon }_r}+\frac{k^2{h}^2}{12}+\frac{h^2}{6}\frac{1}{{\epsilon }_r}{\lambda }_i\right)a_{N+1}$.

Otherwise, multiplying $C_x^{-1}$ on both sides of Equation (33), we have

$$\begin{array}{cc}\hfill & 2T{L}_1{C}_x{\overline{u}}_{:,N}+\left(2h\eta I_{M+2}+T{L}_2\right)C_x{\overline{u}}_{:,N+1}\hfill \\ \hfill =& 2h\eta g+T\left(h^2{F}_{:,N+1}+\frac{h^4}{12}\Delta F_{:,N+1}-\eta \frac{h^3}{3}{(f_y)}_{:,N+1}\right).\hfill \end{array}$$

(36)

Combing (35) and (36), it follows

$$\begin{array}{cc}\hfill & \left(2h\eta C_x+T{L}_2{C}_x-2T{L}_1{C}_x{P}^{-1}Q\right){\overline{u}}_{:,N+1}\hfill \\ \hfill =& 2h\eta g+T\left(h^2{F}_{:,N+1}+\frac{h^4}{12}\Delta F_{:,N+1}-\eta \frac{h^3}{3}{(f_y)}_{:,N+1}\right)-2T{C}_x{\mathbb{L}}_i^{-1}{P}^{-1}{\overline{F}}_{:,N}.\hfill \end{array}$$

(37)

4. Fast High-Order Schemes for the Scattering from Finite Array of Cavities

In this section, we present fast algorithms to solve the electromagnetic scattering from the finite array of large cavities in the TE case. Consider the electromagnetic scattering from the finite array of large cavities ${\mathsf{\Omega }}_1,{\mathsf{\Omega }}_2,\dots ,{\mathsf{\Omega }}_n$ embedded in an infinite ground plane. The geometry of the finite array of large cavities is shown in Figure 2. Assume the walls $S_l,l=1,2,\dots ,n$ of the cavities and the ground plane ${\mathsf{\Gamma }}_c$ are perfect electric conductors. ${\mathsf{\Gamma }}_l$ is the aperture of the cavity ${\mathsf{\Omega }}_l$ separately. The half-space above the ground plane is filled with a homogeneous medium characterized by its dielectric permittivity ${\epsilon }_0$ and magnetic permeability ${\mu }_0$. The interior of the finite array of large cavities is filled with the nonmagnetic material with the relative dielectric permittivity ${\epsilon }_r^{(l)}$ respectively.

Figure 2. The geometric model of the finite array of cavities.

For the TE polarization case, Maxwell equations are reduced to the Helmholtz equations with Neumann boundary conditions. To reduce the scattering problem into bounded domain ${\mathsf{\Omega }}_l,l=1,2,\dots ,n$, we need to obtain the boundary conditions on ${\mathsf{\Gamma }}_l$. Rewrite the Helmholtz equation as

$$\begin{array}{cccc}\hfill & \frac{1}{{\epsilon }_r^{(l)}}\Delta u^{(l)}+k_0^2{u}^{(l)}=0,\hfill & \hfill \mathrm{in}& {\mathsf{\Omega }}_l,\hfill \\ \hfill & \frac{1}{{\epsilon }_r^{(l)}}\frac{\partial u^{(l)}}{\partial n}=0,\hfill & \hfill \mathrm{on}& S_l\cup {\mathsf{\Gamma }}^c.\hfill \end{array}$$

(38)

Based on Equation (14), it follows that

$$u=T\frac{1}{{\epsilon }_r}\left(\frac{\partial u}{\partial n}\right)+g(x),\mathrm{on}\bigcup _{l=1}^n{\mathsf{\Gamma }}_j.$$

(39)

Extend $\frac{\partial u^{(l)}}{\partial n}$ to the whole x-axis by

$$\frac{\partial {\tilde{u}}^{(l)}}{\partial n}=\left\{\begin{array}{cc}\frac{\partial u^{(l)}}{\partial n},\hfill & \hfill x\in {\mathsf{\Gamma }}_l,\\ 0,\hfill & \hfill x\in \mathbb{R}\backslash {\mathsf{\Gamma }}_l.\end{array}\right.$$

For the total magnetic field $\frac{\partial u}{\partial \mathrm{n}}$, we have its x-axis extension

$$\frac{\partial \tilde{u}}{\partial n}=\left\{\begin{array}{cc}\frac{\partial u^{(l)}}{\partial n},\hfill & \hfill x\in {\mathsf{\Gamma }}_l,\\ 0,\hfill & \hfill x\in {\mathsf{\Gamma }}^c.\end{array}\right.$$

We have the following relationship

$$\tilde{u}={\tilde{u}}^{(1)}+{\tilde{u}}^{(2)}+\cdots +{\tilde{u}}^{(l)}+g^{(l)},\mathrm{on}\mathsf{\Gamma }\cup {\mathsf{\Gamma }}^c,$$

where $g^{(l)}=2e^{\mathrm{i}\alpha x^{(l)}}$. The transparent boundary condition (39) can be rewritten as

$$\tilde{u}=T\frac{1}{{\epsilon }_r^{(1)}}\left(\frac{\partial {\tilde{u}}^{(1)}}{\partial n}\right)+T\frac{1}{{\epsilon }_r^{(2)}}\left(\frac{\partial {\tilde{u}}^{(2)}}{\partial n}\right)+\cdots +T\frac{1}{{\epsilon }_r^{(n)}}\left(\frac{\partial {\tilde{u}}^{(n)}}{\partial n}\right)+g^{(l)}.$$

(40)

Therefore, the electromagnetic scattering from the finite array of large cavities in the TE polarization can be described as follows,

$$\begin{array}{cccc}\hfill & \frac{1}{{\epsilon }_r^{(l)}}\Delta u^{(l)}+k_0^2{u}^{(l)}=0,\hfill & \hfill \mathrm{in}& {\mathsf{\Omega }}_l,\hfill \\ \hfill & \frac{1}{{\epsilon }_r^{(l)}}\frac{\partial u^{(l)}}{\partial n}=0,\hfill & \hfill \mathrm{on}& S_l\cup {\mathsf{\Gamma }}^c,\hfill \\ \hfill & u^{(l)}=T\frac{1}{{\epsilon }_r^{(l)}}\left(\frac{\partial {\tilde{u}}^{(l)}}{\partial n}\right)+\sum _{\begin{array}{c}i=1\\ i\ne l\end{array}}^{n}T\frac{1}{{\epsilon }_r^{(i)}}\left(\frac{\partial {\tilde{u}}^{(i)}}{\partial n}\right)+g^{(l)},\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_l.\hfill \end{array}$$

(41)

The Fast Algorithm for the Scattering by a Finite Array of Cavities

Consider the second-order discretization of the transparent boundary condition on $\mathsf{\Gamma }$

$$u_{:,N+1}^{(l)}=\frac{1}{{\epsilon }_r^{(l)}}T\frac{u_{:,N+2}^{(l)}-u_{:,N}^{(l)}}{2h}+\sum _{j=1,j\ne l}^n\frac{1}{{\epsilon }_r^{(j)}}T\frac{u_{:,N+2}^{(j)}-u_{:,N}^{(j)}}{2h}+g^{(l)},\mathrm{on}{\mathsf{\Gamma }}_l.$$

(42)

We rewrite (42) as matrix form

$$2h{u}_{:,N+1}^{(l)}-\frac{1}{{\epsilon }_r^{(l)}}T\left(u_{:,N+2}^{(l)}-u_{:,N}^{(l)}\right)-\sum _{j=1,j\ne l}^n\frac{1}{{\epsilon }_r^{(j)}}T\left(u_{:,N+2}^{(j)}-u_{:,N}^{(j)}\right)=2h{g}^{(l)},\mathrm{on}{\mathsf{\Gamma }}_l.$$

(43)

On each aperture ${\mathsf{\Gamma }}^{(l)}$, we have the following second-order approximation according to Equation (20)

$$\begin{array}{cc}\hfill & \left(\frac{1}{{\epsilon }_r^{(l)}}\left(A_x-\frac{2}{h^2}{I}_{M+2}\right)+k_0^2\right)u_{:,N+1}^{(l)}+\frac{1}{{\epsilon }_r^{(l)}}\frac{1}{h^2}{u}_{:,N}^{(l)}+\frac{1}{{\epsilon }_r^{(l)}}\frac{1}{h^2}{u}_{:,N+2}^{(l)}\hfill \\ \hfill =& F_{:,N+1}^{(l)},l=1,2,\dots ,n.\hfill \end{array}$$

(44)

Multiplying $C_x^{-1}$ on both sides of (44), we have

$$\left(\frac{1}{{\epsilon }_r^{(l)}}\left(\mathsf{\Lambda }-\frac{2}{h^2}{I}_{M+2}\right)+k_0^2\right){\overline{u}}_{:,N+1}^{(l)}+\frac{1}{{\epsilon }_r^{(l)}}\frac{1}{h^2}{\overline{u}}_{:,N}^{(l)}+\frac{1}{{\epsilon }_r^{(l)}}\frac{1}{h^2}{\overline{u}}_{:,N+2}^{(l)}={\overline{F}}_{:,N+1}^{(l)}.$$

(45)

Denoting ${\displaystyle J_1^{(l)}=\frac{1}{{\epsilon }_r^{(l)}}\left(\mathsf{\Lambda }-\frac{2}{h^2}{I}_{M+2}\right)+k_0^2}$, ${\displaystyle J_2^{(l)}=\frac{1}{{\epsilon }_r^{(l)}}\frac{1}{h^2}}$, we have

$$\begin{array}{cc}\hfill & \left(2h{C}_x+\frac{1}{{\epsilon }_r^{(l)}}T{C}_x{J}_2^{{(l)}^{-1}}{J}_1^{(l)}\right){\overline{u}}_{:,N+1}^{(l)}+2\frac{1}{{\epsilon }_r^{(l)}}T{C}_x{\overline{u}}_{:,N}^{(l)}\hfill \\ \hfill & +\sum _{j=1,j\ne l}^n\left(T\frac{1}{{\epsilon }_r^{(j)}}{C}_x{J}_2^{{(j)}^{-1}}{J}_1^{(j)}{\overline{u}}_{:,N+1}^{(j)}+2\frac{1}{{\epsilon }_r^{(j)}}T{C}_x{\overline{u}}_{:,N}^{(j)}\right)\hfill \\ \hfill =& 2h{g}^{(l)}+\sum _{j=1}^{n}T{C}_x{J}_2^{{(j)}^{-1}}{F}_{:,N+1}^{(j)}.\hfill \end{array}$$

(46)

We apply the fast algorithm in each cavity ${\mathsf{\Omega }}_l$, and obtain

$$P^{(l)}{\overline{u}}_{:,N}^{(l)}+Q^{(l)}{\overline{u}}_{:,N+1}^{(l)}={\tilde{F}}^{(l)},l=1,2,\dots ,n,$$

(47)

where

$$\begin{array}{cc}\hfill & P^{(l)}=\mathrm{diag}\left(p_1^{(l)},p_2^{(l)},\dots ,p_{M+2}^{(l)}\right),Q^{(l)}=\mathrm{diag}\left(q_1^{(l)},q_2^{(l)},\dots ,q_{M+2}^{(l)}\right),\hfill \\ & {\tilde{F}}^{(l)}={\left({\tilde{f}}_{0,N}^{(l)},{\tilde{f}}_{1,N}^{(l)},\dots ,{\tilde{f}}_{M+1,N}^{(l)}\right)}^T.\hfill \end{array}$$

Combining (46) and (47) yields

$$\begin{array}{cc}\hfill & \left(2h{C}_x+\frac{1}{{\epsilon }_r^{(l)}}T{C}_x{J}_2^{{(l)}^{-1}}{J}_1^{(l)}-2\frac{1}{{\epsilon }_r^{(l)}}T{C}_x{P}^{{(l)}^{-1}}{Q}^{(l)}\right){\overline{u}}_{:,N+1}^{(l)}\hfill \\ \hfill & +\sum _{j=1,j\ne l}^n\left(\frac{1}{{\epsilon }_r^{(j)}}T{C}_x{J}_2^{{(j)}^{-1}}{J}_1^{(j)}-2\frac{1}{{\epsilon }_r^{(j)}}T{C}_x{P}^{{(j)}^{-1}}{Q}^{(j)}\right){\overline{u}}_{:,N+1}^{(j)}\hfill \\ \hfill =& 2h{g}^{(l)}+\sum _{j=1}^n\frac{1}{{\epsilon }_r^{(j)}}T\left(J_2^{{(j)}^{-1}}{\overline{F}}_{:,N+1}^{(j)}+2C_x{P}^{{(j)}^{-1}}{\tilde{F}}^{(j)}\right).\hfill \end{array}$$

(48)

Therefore, we can derive the global system for the scattering problem from the finite array of large cavities in TE polarization

$$\left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1n}\\ A_{21}& A_{22}& \cdots & A_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{n1}& A_{n2}& \cdots & A_{nn}\end{array}\right)\left(\begin{array}{c}{\overline{u}}_{:,N+1}^{(1)}\\ {\overline{u}}_{:,N+1}^{(2)}\\ ⋮\\ {\overline{u}}_{:,N+1}^{(n)}\end{array}\right)=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right),$$

(49)

where

$$\begin{array}{cc}\hfill & A_{ii}=2h{C}_x+\frac{1}{{\epsilon }_r^{(i)}}T{C}_x{J}_2^{{(i)}^{-1}}{J}_1^{(i)}-2\frac{1}{{\epsilon }_r^{(i)}}T{C}_x{P}^{{(i)}^{-1}}{Q}^{(i)},\hfill \\ \hfill & A_{ij}=\frac{1}{{\epsilon }_r^{(j)}}T{C}_x{J}_2^{{(j)}^{-1}}{J}_1^{(j)}-2T\frac{1}{{\epsilon }_r^{(j)}}{C}_x{P}^{{(j)}^{-1}}{Q}^{(j)},\hfill \\ \hfill & b_i=2h{g}^{(i)}+\sum _{j=1}^n\frac{1}{{\epsilon }_r^{(j)}}T{C}_x\left(J_2^{{(j)}^{-1}}{\overline{F}}_{:,N+1}^{(j)}+2P^{{(j)}^{-1}}{\tilde{F}}^{(j)}\right),\hfill \\ \hfill & i=1,2,\dots ,n,j=1,2,\dots ,n.\hfill \end{array}$$

Similarly, we can apply the fast algorithm for the fourth-order scheme. Then we can derive the final fourth-order system on the apertures, which has the same expression as (49), where

$$\begin{array}{cc}\hfill & A_{ii}=\left(2h\eta +T{L}_2^{(i)}\right)C_x-2T{L}_1^{(i)}{C}_x{(P^{(i)})}^{-1}{Q}^{(i)},\hfill \\ \hfill & A_{ij}=T{L}_2^{(i)}{C}_x-2T{L}_1^{(i)}{C}_x{(P^{(j)})}^{-1}{Q}^{(j)},\hfill \\ \hfill & b_i=2h\eta g^{(i)}+\sum _{j=1}^{n}T\left(h^2{F}_{:,N+1}^{(j)}+\frac{h^4}{12}\Delta F_{:,N+1}^{(j)}-\eta \frac{h^3}{3}{(f_y)}_{:,N+1}^{(j)}\right)\hfill \\ \hfill & -\sum _{j=1}^{n}2T{C}_x{(P^{(j)})}^{-1}{({\mathbb{L}}_i^{(i)})}^{-1}{\overline{F}}_{:,N}^{(j)},i=1,2,\dots ,n,j=1,2,\dots ,n.\hfill \end{array}$$

5. Numerical Experiments

In this section, a variety of numerical experiments are demonstrated to examine the efficiency of the proposed fast high-order algorithms for electromagnetic scattering by the finite array of cavities with high wave numbers. All the examples are computed with double precision and are performed on a desktop with an Intel(R) Core(TM) i5-9400F CPU, 2.90 GHz, and 16.00 GB memory. We present errors in $L^{\infty }$ norm and estimate the convergence order by

$$r=\frac{log(\parallel e_{h_1}{\parallel }_{\infty }/\parallel e_{h_2}{\parallel }_{\infty })}{log(h_1/h_2)}.$$

5.1. Example 1

We consider a rectangular cavity of 1 m wide and 1 m deep. The exact solution is selected as

$$u=cos(k_{0}x)sin((k_0+\pi /2)y),(x,y)\in [0,1]\times [-1,0].$$

The source term $f(x)$ is derived from the (14) by changing the wave number and the relative permittivity. The errors and the convergence rates for the cavity filled with different materials are given in Table 1, Table 2 and Table 3. It is shown that the convergence orders for the schemes are two and four respectively. The numerical solution and the error of the proposed schemes for Example 1 with $k_0=16\pi$m${}^{-1}$, $h=1/1024$ m are represented in Figure 3.

Table 1. Numerical convergence orders of the second-order fast algorithm with $k_0=8\pi$m${}^{-1}$$,{\epsilon }_r=4+\mathrm{i}$.

h	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Gamma })$	Order	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Omega })$	Order	CPU Time (s)
1/16	2.0281 × ${10}^{-1}$		9.7386 × ${10}^{-2}$		0.0080
1/32	4.6314 × ${10}^{-2}$	2.1306	4.4007 × ${10}^{-2}$	1.1460	0.0982
1/64	1.1937 × ${10}^{-2}$	1.9560	1.3852 × ${10}^{-2}$	1.6676	0.0200
1/128	3.0142 × ${10}^{-3}$	1.9856	3.4765 × ${10}^{-3}$	1.9944	0.0317
1/256	7.5720 × ${10}^{-4}$	1.9931	8.7693 × ${10}^{-4}$	1.9871	0.0894

Table 2. Numerical convergence orders of the fourth-order fast algorithm with $k_0=8\pi$m${}^{-1}$$,{\epsilon }_r=4+\mathrm{i}$.

h	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Gamma })$	Order	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Omega })$	Order	CPU Time (s)
1/16	2.9375 × ${10}^{-1}$		1.6377 × ${10}^{-1}$		0.0382
1/32	1.9114 × ${10}^{-2}$	3.9419	1.1956 × ${10}^{-2}$	3.7758	0.0561
1/64	1.3079 × ${10}^{-3}$	3.8694	1.1329 × ${10}^{-3}$	3.3997	0.0737
1/128	8.4704 × ${10}^{-5}$	3.9486	8.0131 × ${10}^{-5}$	3.8215	0.1400
1/256	5.3553 × ${10}^{-6}$	3.9834	5.2272 × ${10}^{-6}$	3.9382	0.6564

Table 3. Numerical convergence orders of the fourth-order fast algorithm with $k_0=16\pi$m${}^{-1}$$,{\epsilon }_r=4+\mathrm{i}$.

h	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Gamma })$	Order	${\mathit{e}}_{\mathbf{\infty }}(\mathsf{\Omega })$	Order	CPU Time (s)
1/64	2.1762 × ${10}^{-2}$		1.4052 × ${10}^{-2}$		0.0478
1/128	1.4300 × ${10}^{-3}$	3.9277	1.2347 × ${10}^{-3}$	3.5086	0.0908
1/256	9.1459 × ${10}^{-5}$	3.9668	8.6201 × ${10}^{-5}$	3.8403	0.5833
1/512	5.7521 × ${10}^{-6}$	3.9910	5.6021 × ${10}^{-6}$	3.9437	5.4535
1/1024	3.6003 × ${10}^{-7}$	3.9979	3.5557 × ${10}^{-7}$	3.9778	66.1697

Figure 3. The real part of the numerical solution (left) and the absolute error plot (right) for the empty cavity with $k_0=16\pi$m${}^{-1}$, $h=1/1024$ m.

5.2. Example 2

We examine the validity of the proposed algorithms for the scattering problem from the single cavity further. We consider a 1 m wide and 0.25 m deep cavity scattering illuminated by a 300-MHz TE plane wave. The example was tested by Du’s results [30]. The physical parameter backscatter RCS can be calculated as follows. Let $\sigma (\theta )=\frac{4}{k_0}{\left|P(\theta )\right|}^2$, where $\theta$ is the observation angle and P is the far-field coefficient given by

$$P(\theta )=\frac{k_0}{2}sin\theta {\int }_{\mathsf{\Gamma }}{\mathrm{e}}^{\left(\mathrm{i}{k}_{0}xcos\theta \right)}udx.$$

When the incident angle is the same as the observation direction, the backscatter RCS is defined by

$$\mathrm{Backscatter}\mathrm{RCS}(\theta )=10{log}_{10}\sigma (\theta )dB.$$

Figure 4 and Figure 5 display the magnitude and RCS of cavities filled with the homogeneous media ${\epsilon }_r=1$ and ${\epsilon }_r=4+\mathrm{i}$ in TE polarization, where ‘o’ denotes the solutions derived in [30]. Clearly, the numerical results coincide with those of Du’s method. Moreover, we can see from Table 4 that the fast algorithm remarkably reduces the computational time.

Figure 4. The magnitude of the aperture fields at normal incidence (left) and backscatter RCS (right) in TE polarization with $k_0=2\pi$m${}^{-1}$, ${\epsilon }_r=1,h=1/256$ m, ‘o’: the solutions derived in [30], ‘-’: our results.

Figure 5. The magnitude of the aperture fields at normal incidence (left) and backscatter RCS (right) in TE polarization with $k_0=2\pi$m${}^{-1}$, ${\epsilon }_r=4+\mathrm{i},h=1/256$ m, ‘o’: the solutions derived in [30], ‘-’: our results.

Table 4. The comparison of the computational time for the proposed normal and fast fourth-order scheme.

h	CPU Time of Fourth-Order Scheme (s)	CPU Time of Fast Fourth-Order Scheme (s)
1/128	0.0211	0.0083
1/256	0.3503	0.0544
1/512	1.3928	0.2142
1/1024	8.1006	1.2517
1/2048	out of memory	10.7882

5.3. Example 3

In order to address the electromagnetic scattering from the finite array of large cavities, we first consider a two-cavity scattering problem by two identical rectangular cavities of 1 m wide and 1 m deep. It is investigated that the farther the distance between the finite array of cavities, the smaller the coupling impact of each other, see [31] for details. Therefore, for high wave number problems, we consider the distance between cavities is equal to the width of the cavity. Backscatter RCS comparisons for the scattering from two cavities in TE polarization are displayed in Figure 6 and Figure 7. The cavities are filled with ${\epsilon }_r=[{\epsilon }_r^{(1)},{\epsilon }_r^{(2)}]=[2,2]$, complex material ${\epsilon }_r=[{\epsilon }_r^{(1)},{\epsilon }_r^{(2)}]=[2+0.1\mathrm{i},2+0.1\mathrm{i}]$, and ${\epsilon }_r=[{\epsilon }_r^{(1)},{\epsilon }_r^{(2)}]=[2+10\mathrm{i},2+10\mathrm{i}]$ respectively. We can see that the backscatter RCS of cavities filled with a lossy medium is lower than that of cavities filled with a lossless medium. Moreover, it is observed that the backscatter RCS of cavities decreases with the increase of the imaginary part of the medium. Therefore, the RCS can be reduced by filling the cavity with the absorbing material coating.

Figure 6. The backscatter RCS by two cavities in TE polarization with $k_0=4\pi$m${}^{-1}$ on $256\times 256$ meshes for the media $[2,2]$, $[2+0.1\mathrm{i},2+0.1\mathrm{i}]$ and $[2+10\mathrm{i},2+10\mathrm{i}]$.

Figure 7. The backscatter RCS by two cavities in TE polarization with $k_0=8\pi$$m^{-1}$ on $512\times 512$ meshes for the media $[2,2]$, $[2+0.1\mathrm{i},2+0.1\mathrm{i}]$ and $[2+10\mathrm{i},2+10\mathrm{i}]$.

5.4. Example 4

We consider a three-cavity scattering problem by three identical rectangular cavities that are 1 meter wide and 0.5 meters deep. The distance between cavities is equal to the width of the cavity. We plot the contour for the real part of the magnetic field with $k_0=8\pi$m${}^{-1}$ and $k_0=16\pi$m${}^{-1}$ for the media of ${\epsilon }_r=2.38+0.12\mathrm{i}$ at $\theta =\pi /3$ and $\theta =\pi /4$ in Figure 8 and Figure 9.

Figure 8. Contour for real part of the magnetic field by three empty cavities with $k_0=8\pi$m${}^{-1}$ of $512\times 256$ mesh at $\theta =\pi /3$.

Figure 9. The real part of the magnetic field by three empty cavities with $k_0=16\pi$m${}^{-1}$ of $1024\times 512$ mesh at $\theta =\pi /4$.

6. Conclusions

In this paper, we present fast high-order algorithms for solving the electromagnetic scattering from the finite array of cavities with large wave numbers in TE polarization. Compact second-order and fourth-order schemes are applied to discretize the magnetic field equation in TE polarization. The boundary conditions of the cavity arrays are decoupled by a new transparent boundary condition. The fast algorithm is based on the FFT-cosine transformation in the horizontal direction and the Gaussian elimination in the vertical direction. Numerical experiments validate the convergence order and the efficiency of the proposed fast high-order algorithms.