High Order Compact Finite Difference Schemes for the Helmholtz Equation with Discontinuous Coefficients

Feng, Xiufang; Li, Zhilin; Qiao, Zhonghua

doi:10.4208/jcm.1010-m3204

Cited by 36 publications

(21 citation statements)

References 21 publications

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“…If the second order derivatives of the solution have jumps at the interfaces of the layered media, both the second order and fourth order scheme will only result in the first order accuracy. The Helmholtz interior problem with this kind of solution has been considered in [26].…”

Section: Examplementioning

confidence: 99%

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

Zhao

Qiao

Tang

2011

JCM

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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.

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Section: Examplementioning

confidence: 99%

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

Zhao

Qiao

Tang

2011

JCM

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“…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.…”

Section: Discussionmentioning

confidence: 99%

Electromagnetic scattering from a cavity embedded in an impedance ground plane

Sun

et al. 2018

Math Methods in App Sciences

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This paper is concerned with time‐harmonic electromagnetic scattering from a cavity embedded in an impedance ground plane. The fillings (which may be inhomogeneous) do not protrude the cavity and the space above the ground plane is empty. This problem is obviously different from those considered in previous work where either perfectly conducting boundary conditions were used or the cavity was assumed to be empty. By employing the Green's function method, we reduce the scattering problem to a boundary‐value problem in a bounded domain (the cavity), with impedance boundary conditions on the cavity walls and an impedance‐to‐Dirichlet condition on the cavity aperture. Existence and uniqueness of the solution are proved for the weak formulation of the reduced problem. We also propose a numerical method to calculate the radar cross section (RCS), which is a parameter of physical interest. Numerical experiments show that the proposed model and numerical method are efficient for the calculation of RCS from cavities.

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“…After deriving G q by (10), the solution G of the threedimensional Helmholtz equation is calculated by G R # E R & E F ' G q . Due to the special form of the coefficient matrix, the computational procedure is independent for each , 6 0,1, … , 7 .…”

Section: Parallel Implementationmentioning

confidence: 99%

“…In addition, finite difference method is not competitive for solving the Helmholtz equation with large wave number if no fine mesh provided. Therefore, some high order finite difference method were proposed for solving the twodimensional Helmholtz equation in [5][6][7][8][9][10]. This method is prevalent since they can offer a high accuracy solutions with less computational cost.…”

Section: Introductionmentioning

confidence: 99%

A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary

An¹

2018

ACM

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Helmholtz equation is widely applied in the scientific and engineering problem. For the solution of the threedimensional Helmholtz equation, the computational efficiency of the algorithm is especially important. In this paper, in order to solve the contradiction between accuracy and efficiency, a fast high order finite difference method is proposed for solving the three-dimensional Helmholtz equation. First, a traditional fourth order method is constructed. Then, fast Fourier transformation are introduced to generate a block-tridiagonal structure which can easily divide the original problem into small and independent subsystems. For large 3D problems, the computation of traditional discrete Fourier transformation is less efficient, and the memory requirements increase rapidly. To fix this problem, the transformed coefficient matrix is constructed as a sparse structure. In light of the sparsity, the algorithm presented in this paper requires less memory space and computational cost. This sparse structure also leads to independent solving procedure of any plane in the domain. Therefore, parallel method can be used to solve the Helmholtz equation with large grid number. In the end, three numerical experiments are presented to verify the effectiveness of the fast fourth-order algorithm, and the acceleration effect to use the parallel method has been demonstrated.

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High Order Compact Finite Difference Schemes for the Helmholtz Equation with Discontinuous Coefficients

Cited by 36 publications

References 21 publications

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

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

Electromagnetic scattering from a cavity embedded in an impedance ground plane

A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary

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