This paper addresses nonphysical reflections encountered in the discretization of wave equations on nonuniform grids. Such nonphysical solutions are commonly attributed to spurious modes in the numerical dispersion relation. We provide an example of a discretization in which a (nonspurious) physical mode is spuriously energized at a grid nonuniformity. It is found that, in particular, the box scheme and, in general, certain discretizations within the class of Runge-Kutta methods, are free from such spurious reflections.

Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Frank, J., & Reich, S. (2004). On spurious reflections, nonuniform grids and finite difference discretizations of wave equations. Modelling, Analysis and Simulation [MAS]. CWI.