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.