A Fast High Order Method for Electromagnetic Scattering by Large Open Cavities

Abstract

In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from the open cavity is reduced to a bounded domain problem. A compact fourth order finite difference scheme is then proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach O(h 4 ) in the whole computational domain. A fast algorithm, exploiting the discrete Fourier transformation in the horizontal and a Gaussian elimination in the vertical direction, is employed, which reduces the discrete system to a much smaller interface system. An effective preconditioner is presented for the BICGstab iterative solver to solve this interface system. Numerical results demonstrate the remarkable accuracy and efficiency of the proposed method. In particular, it can be used to solve the cavity model for the large wave number up to 600π.Mathematics subject classification: 65N06, 78M20.

Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

“…In this problem, Ω := (0, 1)×(0, 1 4 ), 1 4 ], Γ t := [1, 0] × { 1 4 }, Γ l := {0} × [ 1 4 , 0], f = 0, u = 0 on Γ b ∪ Γ r ∪ Γ l , ∂u ∂y + iku = g t on Γ t , which is the lowest-order approximation of the radiation boundary condition (see [6,18]). Setting g t = -2ik cos θ e ik sin θx and θ = π 4 (see [6,36]), we show that the real part, the image part and magnitude of the solution with k = 128π, N = 512 in Figs. 3-4, which is consistent with that illustrated in [6,36].…”

“…But compared with the second-and fourth-order schemes and the parameter one, the sixthorder method investigated here can achieve the best computational accuracy in all tested cases. Finally, we consider a practical model which is reduced from the large cavity electromagnetic scattering and has been investigated in [6,[34][35][36]. In this problem, Ω := (0, 1)×(0, 1 4 ), 1 4 ], Γ t := [1, 0] × { 1 4 }, Γ l := {0} × [ 1 4 , 0], f = 0, u = 0 on Γ b ∪ Γ r ∪ Γ l , ∂u ∂y + iku = g t on Γ t , which is the lowest-order approximation of the radiation boundary condition (see [6,18]).…”

“…Further research can be found in [14]. But in many cases, such as the Helmholtz equation after reduction from the large cavity electromagnetic scattering, the inhomogeneous Robin boundary condition (2) is necessary (see [6,[34][35][36]). Obviously, the high-order scheme in this case cannot be got following the process in [12] due to the existence of the function g. On the other hand, the Robin boundary condition is only imposed on a single boundary in [12,14].…”

Electromagnetic scattering from a cavity embedded in an impedance ground plane

“…It is well known that high-order methods are more attractive for solving Helmholtz problems with large wave numbers because they can offer relative higher accurate solutions by utilizing fewer mesh points; eg, see other works. 21,[57][58][59][60][61][62][63] Efficient high-order methods and the corresponding fast algorithms for large wave number cavity problems with impedance boundary conditions are being considered.…”

A Survey of Open Cavity Scattering Problems

“…LI nature of the large cavity problem, which is a perfect example for the long-standing high frequency scattering problems. Tremendous effort was made to develop various fast and accurate numerical methods to solve the large cavity problem [1,11,17,18,26,27,40,41,44,45]. The challenging mathematical issue is to establish the stability estimates with explicit dependence on the high wavenumber [12,13,27], which help us gain a deeper understanding on high frequency problems.…”