Anomalous numerical dispersion is an issue arising out of limited computational resources and the time-domain method currently in use.  The effect could easily be traced back to Nyquist sampling of waveforms for accurate representation.  If `enough'' samples are present, then a waveform could be regenerated from the samples.  `Enough'' could be qualified as at least two sample points per wavelength of the signal being sampled.  In practice, more samples would provide a more accurate sampling from which a more accurate regeneration would be possible.
A 2D FDTD animation of the electric field with 256x256 cells of 0.03 meters per cell side exhibiting numerical dispersion. Note in the center and the lower right corner the faint ripples. Here the 2D electric field is portrayed as white-orange-yellow-green positive with red-purple-blue negative.
static_ds_0p03_256x256

As an example, a 2D FDTD code was written to propagate a pulse which could reflect off a boundary.  As a test case, a sample problem of cell size 0.03m by 0.03m with a total mesh size of 256x256 was created.  This lead to a physical problem size of 256 times 0.03m per side of the mesh of 7.68m.  As can be seen in the animation, the pulse begins to propagate fairly well and encounters the PEC wall at the left of the mesh.

The boundaries are a combination of second-order Mur boundaries along the walls and first-order Mur boundaries at the corners.  Notice that some reflections occur.  The magnitudes of the reflected electric field do appear higher at the corners and when the incident energy is non-normal to the wall.  The reflections are quite low when the incident waveform is normal to the walls.  As the animations continue, the scale of the plot is dynamically shifted to the largest electric field magnitude (either positive or negative).  Since the excitation was a static pulse of electric field, no further energy should enter the system after time 0.0s.  Given a constant in energy and a spreading wave front, one would expect the electric and magnetic field amplitudes to diminish.  The overall energy should remain exactly the same until the wave front begins to exit the mesh with the Mur absorbing boundary conditions.  Once the pulse encounters the PEC barrier in the left side of the mesh, the animation has begun to exhibit a `ringing'' phenomena behind the original wave front.  This could be called anomalous numerical dispersion, numerical dispersion, phase distortion or one of several other bad names.  The overall idea, though, is that it is NOT real in the system under study.  One way to confirm the dispersion is NOT real is to decrease the size of the cell.  In order to model the same physical size system, the overall mesh must be increased accordingly. 

A 2D FDTD animation of the electric field with 768x768 cells of 0.01 meters per cell side exhibiting reduced numerical dispersion. Note in the center and the lower right corner the diminished faint ripples. Here the 2D electric field is portrayed as white-orange-yellow-green positive with red-purple-blue negative.
static_ds_0p01_768x768

In this example, the cell size was decreased from 0.03 to 0.01 meters which increased the overall mesh size to 768x768 cells.  Compare this animation to the previous at 0.03 meter cell size and notice that the dispersion has been alleviated.    It is important in that the issue has been alleviated but not removed.  This could be considered a `failure mode'' of this 2D FDTD implementation and method.  In other words, the simulation did not fail exactly, but the solution will degrade as the physical cell size increases and the solution will improve as the cell size decreases.  One of the beneficial aspects of time-domain local-operator methods is the ability to run in a `degraded'' mode to test boundary conditions and initial problem setup.  Once these issues have been confirmed suitably correct, a much higher resolution (smaller cell size and larger overall mesh size) simulation may be run.  This aids in rapid debugging of user input and code modifications.

Animations in mp4 format: Left) 256x256, Right 768x768
static_ds_0p03_256x256static_ds_0p01_768x768