Research ArticleOpen AccessGoogle Scholar indexed
A Fast Fourth-Order Method for 3D Helmholtz Equation with Neumann Boundary
School of Mathematics and Physics, North China Electric Power University Department Name of Organization, Baoding, China
School of Mathematics and Physics, North China Electric Power University Department Name of Organization, Baoding, China
- 1 School of Mathematics and Physics, North China Electric Power University Department Name of Organization, Baoding, China
- 2 School of Mathematics and Physics, North China Electric Power University Department Name of Organization, Baoding, China
American Journal of Computational Mathematics·Volume 08 (2018)·Pages 222–232·Published 31 August 2018·DOI10.4236/ajcm.2018.83018
Copy link · social · email
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 ( M 3 ). 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.
KeywordsHelmholtz EquationFourier TransformNeumann Boundary Condition
- Singer, I. and Turkel, E. (1998) High-Order Finite Difference Methods for the Helmholtz Equation. Computer Methods in Applied Mechanics & Engineering, 163, 343-358. https://doi.org/10.1016/S0045-7825(98)00023-1
- Harari, I. and Hughes, T.J.R. (1991) Finite Element Methods for the Helmholtz Equation in an Exterior Domain: Model Problems. Computer Methods in Applied Mechanics & Engineering, 87, 59-96. https://doi.org/10.1016/0045-7825(91)90146-W
- Jin, J.M., Liu, J., Lou, Z. and Liang, C.S.T. (2003) A Fully High-Order Finite-Element Simulation of Scattering by Deep Cavities. Antennas & Propagation IEEE Transactions on, 51, 2420-2429. https://doi.org/10.1109/TAP.2003.816354
- Jin, J.M. and Liu, J. (2000) A Special Higher Order Finite-Element Method for Scattering by Deep Cavities. IEEE Transactions on Antennas & Propagation, 48, 694-703. https://doi.org/10.1109/8.855487
- Braverman, E., Israeli, M. and Averbuch, A. (1999) A Fast Spectral Solver for a 3D Helmholtz Equation. Society for Industrial and Applied Mathematics, 20, 2237-2260. https://doi.org/10.1137/S1064827598334241
- Nabavi, M., Siddiqui, M.H.K. and Dargahi, J. (2007) A New 9-Point Sixth-Order Accurate Compact Finite-Difference Method for the Helmholtz Equation. Journal of Sound & Vibration, 307, 972-982. https://doi.org/10.1016/j.jsv.2007.06.070
- Hong, P.L., Minh, T.L. and Hoang, Q.P. (2018) On a Three Dimensional Cauchy Problem for Inhomogeneous Helmholtz Equation Associated with Perturbed Wave Number. Journal of Computational and Applied Mathematics, 335, 86-98. https://doi.org/10.1016/j.cam.2017.11.042
- Hua, Q.S., Gu, Y., Qu, W.Z., Chen, W. and Zhang, C.Z. (2017) A Meshless Generalized Finite Difference Method for Inverse Cauchy Problems Associated with Three-Dimensional Inhomogeneous Helmholtz-Type Equations. Engineering Analysis with Boundary Elements, 82, 162-171. https://doi.org/10.1016/j.enganabound.2017.06.005
- Kashirin, A.A., Smagin, S.I. and Taltykina, M.Yu. (2016) Mosaic-Skeleton Method as Applied to the Numerical Solution of Three-Dimensional Dirichlet Problems for the Helmholtz Equation in Integral Form. Computational Mathematics and Mathematical Physics, 56, 612-625. https://doi.org/10.1134/S0965542516040096
- Britt, S., Tsynkov, S. and Turkel, E. (2010) A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates. Journal of Scientific Computing, 45, 26-47. https://doi.org/10.1007/s10915-010-9348-3