The Triangular Scheme

for a grid defined by points at indices , for integer and such that is even. These coordinates refer to grid points at locations and , so that a given grid point is equidistant from its six neighbors. This arrangement is shown in Figure A.4(a) and can be considered to be a rectilinear grid under a coordinate transformation; we refer to [193] for a discussion of the range of allowable spatial frequencies for such a grid.

In this case, we will again have an amplification polynomial of the form (A.5), with

Because is not multilinear (see §A.2.2) in the cosines, finding the extrema is not as simple as in the interpolated case--one can proceed either through some tedious algebra, change to stretched rectilinear coordinates, in which becomes multilinear again, or make use of a computer. In any case, these extrema can be shown to be

and thus, from (A.9),

for stability |

This is surprising, because the bound for passivity, from (4.80), of the triangular mesh is . That is to say, for a given inter-junction spacing of , a triangular waveguide mesh, of the type mentioned in §4.6.1, is concretely passive for time steps with . The corresponding difference equation, namely (A.18), is

The question which arises here is of the distinction between passive and stable numerical methods (this was also seen for the mesh for the transmission line equations in §4.3.6, as well as in the previous section on the interpolated rectilinear scheme). Is it always possible to find a passive realization of a stable numerical method? The discussion on the hexagonal mesh will help to answer this question. To this end, we note that at the stability limit, we can rewrite as

for a function whose squared magnitude is given by

The spectral amplification factors at the stability limit will then be, from (A.6),

For (its limiting value), the triangular scheme has the same potential for instability as the rectilinear scheme. Linear growth may occur for this scheme at the seven spatial frequency pairs

Here we have taken into account the fact that at the passivity bound, we require one less add per point (in the waveguide mesh implementation, the self-loop disappears). We also mention that the triangular difference scheme is doubly pathological, in the sense that not only do its passivity and stability regimes not coincide (and aside from the interpolated rectilinear schemes, it is the only scheme examined in this appendix that exhibits this behavior), but it also can not be decomposed into even/odd mutually exclusive subschemes, as can all the other schemes to be discussed here (again, excepting the interpolated scheme). It seems reasonable to conjecture that these two ``symptoms'' are related (somehow).