The von Neumann stability analysis [1, 2] discussed in the previous chapter is widely applicable and enables the assessment of the stability of any finite difference scheme in a relatively simple manner. However, such an analysis reveals little about the detailed properties of the difference scheme, and especially the important properties of dispersion and dissipation. These two metrics together yield information about the accuracy of the finite difference algorithm.

In the continuous world, dispersion refers to the variation of the phase velocity vp as a function of frequency or wavelength. Dispersion is present in all materials, although in many cases it can be neglected over a frequency band of interest. Numerical dispersion refers to dispersion that arises due to the discretization process, rather than the physical medium of interest. In addition, the discretization process can lead to anisotropy, where the phase velocity varies with propagation direction; this numerical anisotropy can also be separate from any real anisotropy of the medium.

In addition, as we saw in Figure 5.3, the finite difference scheme can lead to dissipation, which is nonphysical attenuation of the propagating wave. Recall that the example illustrated in Figure 5.3 was in a lossless physical medium, so any dissipation is due only to the discretization process.

In this chapter we will show how dispersion and dissipation arise in the finite difference algorithms that we have discussed so far, and show how to derive the numerical dispersion relation for any algorithm.