A New High-Order Compact Finite Difference Scheme for One-Dimensional Helmholtz Equation using Dirichlet Boundary Conditions

In this paper, tenth-order compact finite difference schemes are developed to solve the Helmholtz equation and are analyzed in one dimension on uniform grids. These schemes are based on sixth-order, eighth-order, and tenth-order approximations to the derivatives calculated from the Helmholtz equation. The schemes are high-order accurate symmetrical representations for Dirichlet boundary conditions. The convergence properties are analyzed and implemented to solve the resulting linear algebraic system. The efficiency, accuracy, and robustness of the schemes are substantiated by their application to test problems having exact solutions. The expected accuracy of the schemes is shown with numerical results. The schemes behave vehemently with respect to the wavenumber.