Short Communication A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation

A new 9-point sixth-order accurate compact finite-difference method for solving the Helmholtz equation in one and two dimensions, is developed and analyzed. This scheme is based on sixth-order approximation to the derivative calculated from the Helmholtz equation. A sixth-order accurate symmetrical representation for the Neumann boundary condition was also developed. The efficiency and accuracy of the scheme is validated by its application to two test problems which have exact solutions. Numerical results show that this sixth-order scheme has the expected accuracy and behaves robustly with respect to the wave number. r 2007 Elsevier Ltd. All rights reserved. The Helmholtz equation, ðD þ k 2 Þu ¼ f , is an elliptic partial differential equation which is a time-harmonic solution of the wave equation. The Helmholtz equation governs some important physical phenomena. These include the potential in time harmonic acoustic and electromagnetic fields, acoustic wave scattering, noise reduction in silencers, water wave propagation, membrane vibration and radar scattering. Obtaining an efficient and more accurate numerical solution for the Helmholtz equation has been the subject of many studies. The numerical solution of the Helmholtz equation has been developed using different approaches such as the finite-difference method (1), the boundary element method (2), the finite-element method (3) and the spectral-element method (4). The boundary element method is derived through the discretization of an integral equation that is mathematically equivalent to the original partial differential equation. The disadvantages of boundary element methods are the restriction to linear problems in homogeneous and isotropic media, as well as the large computer storage space required and lengthy processing time needed to solve the inherent problems encountered with characteristic wave numbers. Finite-element methods are used extensively to solve the Helmholtz equation. In addition to the high- computational cost, another disadvantage of Galerkin finite-element method for solving the Helmholtz