Abstract
Helmholtz equation is widely applied in the scientific and engineering problem. For the solution of the three-dimensional 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.
Highlights
A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet BoundaryTo cite this article: Sheng An, Gendai Gu, Meiling Zhao
In this paper, we consider the following Helmholtz equation ∆,, in Ω, # (1)with Dirichlet boundary condition, on Ω, # (2)where is the wave number
The precise numerical solution of Helmholtz equation is of great importance
Summary
To cite this article: Sheng An, Gendai Gu, Meiling Zhao. A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary. Received: July 26, 2018; Accepted: August 13, 2018; Published: September 11, 2018
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