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https://doi.org/10.1016/0045-7825(91)90146-W

has been cited by the following article:

• TITLE: A Fast Fourth-Order Method for 3D Helmholtz Equation with Neumann Boundary

  AUTHORS: Na Zhu, Meiling Zhao

  KEYWORDS: Helmholtz Equation, Fourier Transform, Neumann Boundary Condition

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.8 No.3, September 12, 2018

  ABSTRACT: We present fast fourth-order finite difference scheme for 3D Helmholtz equation with Neumann boundary condition. We employ the discrete Fourier transform operator and divide the problem into some independent subproblems. By means of the Gaussian elimination in the vertical direction, the problem is reduced into a small system on the top layer of the domain. The procedure for solving the numerical solutions is accelerated by the sparsity of Fourier operator under the space complexity of O(M3). Furthermore, the method makes it possible to solve the 3D Helmholtz equation with large grid number. The accuracy and efficiency of the method are validated by two test examples which have exact solutions.