The Hexagonal Scheme

The hexagonal scheme is different from those previously discussed in
that updating is not the same at every point on the grid. Indeed,
one-half the grid points have a ``mirror-image'' orientation with
respect to the other half, as shown in Figure 5(a). For
this reason, we will take special care in the analysis of this system;
first suppose that we have two grid functions and
defined over the two sub grids (labelled 1 and 2, in Figure
5). We index these two grid functions as
and
, for and integer such that , for
integer , and is even.
will serve as an
approximation to some continuous function at the point
, and
will
approximate a function at a point with coordinates
. As before the distance
between any grid point and its nearest neighbors (three in this case)
is . The difference scheme for the hexagonal waveguide mesh
can then be written as the system

Consistency of (20) with the wave equation is not immediately apparent. We can check it as follows. First expand (20) in a Taylor series in terms of the continuous functions and to get

to . This system can then be reduced to

where is either of or . Discarding higher order terms in and gives the wave equation.

In terms of the spatial Fourier spectra of the grid functions and , we may write the differencing system (20) in the vector form of (10) with

where

Because is Hermitian, we can then change variables so that the system is the form of (11), with

The necessary stability condition, from (12) will then be

It is easy to check that takes on a maximum of 3 when , and is minimized for , and for , , where it takes on the value 0. It is then easy to show that we require in order to satisfy (21). This coincides with the passivity bound, from Eqn. (4.79) of [2].

An analysis of numerical dispersion is more complex in the vector case. Beginning from the uncoupled system defined by
**, whose upper and lower diagonal entries we will call
and
respectively, we can see that we will thus have two pairs of spectral amplification factors, one for each uncoupled scalar equation. These will be given by
**

It is useful to check the values of the amplification factors at the spatial DC frequency, and at the stability bound, where we have , . At this frequency, the spectral amplification factors take on the values

Clearly, the pair of spectral amplification factors correctly represents wave propagation at spatial DC, but the factors will be responsible for

and thus and . Because scheme (20) is consistent with the wave equation, then for any reasonable choice of initial conditions, we must have that , as becomes small. Thus , the component of the numerical solution whose spectral amplification is governed by the parasitic factor must vanish in this limit as well.

The computational and add densities, for the general scheme (20), and at the stability limit for will be given by

As in the rectilinear scheme, we have used the fact that the hexagonal scheme decouples into two independent subschemes at the stability limit.

One other point is worthy of comment. Consider again the vector equation which describes the time evolution of the spatial spectra for the hexagonal scheme, which, in diagonalized form, is exactly (11). At the stability limit, then, for , we will have

Let us examine the second uncoupled subsystem. From (22), the spectral amplification factors will then be

It is of interest to see the effect of the amplification factors after

The important point here is that the

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