We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.
"Modified Finite Difference Schemes on Uniform Grids for Simulations of the Helmholtz Equation at Any Wave Number." J. Appl. Math. 2014 1 - 9, 2014. https://doi.org/10.1155/2014/673106