| Helmholtz equations have a wide range of applications in physics and engineering.Due to its importance in some research fields,a lot of effort has been devoted to researching fast and accurate solutions.At present,the solution of three-dimensional high wave number Helmholtz equation is still a research hotspot.As the wave number increases,the accuracy of the numerical solution decreases.How to improve the accuracy of the numerical solution,shorten the calculation time is the key to effectively solve the high wave number Helmholtz equation.In this paper,a series of high-order compact difference formats and numerical examples are presented for the three-dimensional high wave number Helmholtz equation.Firstly,a fourth-order compact finite difference scheme is constructed,the boundary is discretized in high-order,and the numerical solution method is given.Secondly,in order to further improve the accuracy of numerical solutions and save calculation time,the fourth-order difference scheme is optimized and improved,and sixth-order compact finite difference scheme is established,and the boundary is discretized by sixth-order difference.Then,using the properties of tensor product and discrete Fourier sine transform method,the original problem is divided into independent small systems to solve,which greatly reduces the solving time.Finally,the feasibility and accuracy of the proposed method are verified by a large number of numerical examples,and it is shown that the proposed method can effectively solve the three-dimensional Helmholtz equation problem with Dirichlet and Neumann boundary conditions at high wave number.The method has high precision and good application prospect,and is easy to implement. |
