In this very interesting study the authors examine a number of finite difference stencils for the Helmholtz or reduced wave equation on non-uniform as well as uniform grids in one, two or three dimensions. These schemes are divided into: pointwise, which are the standard finite difference approximations; weighted-average, where the undifferentiated terms are averaged over adjacent grid points; exact-phase representations, where the second difference term is scaled by a parameter designed to eliminate numerical dispersion; and higher-order, based on Padé approximations of the second derivative.
These schemes are analyzed by substituting in plane waves. The authors examine and catalogue dispersion relations, anisotropy (directionally varying dispersion), and spurious reflections caused by grid-size transitions or changes in physical properties. They conclude with general suggestions for enhancing performance of difference schemes.