TITLE:
Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers
AUTHORS:
Chengjie Tong, Xianqi Fang, Meiling Zhao
KEYWORDS:
Impedance Boundary Condition, Helmholtz Equation, Compact Finite Difference, Fourier Sine Transform, Large Wave Numbers
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.13 No.2,
May
6,
2023
ABSTRACT: 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.