Author(s)

Chengjie Tong, Xianqi Fang, Meiling Zhao

School of Mathematics and Physics, North China Electric Power University, Baoding, China.

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in  and -norms, and costs less CPU calculation time and random access memory.

Impedance Boundary Condition, Helmholtz Equation, Compact Finite Difference, Fourier Sine Transform, Large Wave Numbers

Tong, C. , Fang, X. and Zhao, M. (2023) Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers. American Journal of Computational Mathematics, 13, 211-229. doi: 10.4236/ajcm.2023.132011.