High-order finite difference methods for the Helmholtz equation

High-order finite difference methods for solving the Helmholtz equation are developed and analyzed, in one and two dimensions on uniform grids. The standard pointwise representation has a second-order accurate local truncation error. We also study two schemes which have a fourth-order accurate local truncation error. One of the high-order schemes is based on generalizations of the Padé approximation. The second scheme is based on high-order approximation to the derivative calculated from the Helmholtz equation itself. A symmetric high-order representation is developed for a Neumann boundary condition. Numerical results are presented on model problems approximated with the developed schemes.