The (3+1)D Interpolated Rectilinear Scheme

In the interest of achieving a more uniform numerical dispersion profile in (3+1)D, it is of course possible to define an interpolated scheme [155,158], in the same way as was done in (2+1)D in §A.2.2. We will again have a two-step scheme, and updating at a given grid point is performed with reference to, at the previous time step, the grid point at the same location, as well as the 26 nearest neighbors: the six points a distance away, twelve points at a distance of , and eight points that are away--see Figure A.11(a). We present here a complete analysis of the relevant stability conditions, as well as the conditions under which a waveguide mesh implementation exists. We also look at a means of minimizing directional dependence of the numerical dispersion.

Like the cubic rectilinear and octahedral schemes, this scheme will be defined over a rectilinear grid indexed by , and and will have the general form

In order for scheme (A.30) to satisfy the wave equation, we require the constants , , and to satisfy the constraints

and a family of difference schemes parametrized by , and results.

The stability analysis of this scheme proceeds along the same lines as that of the (2+1)D scheme, though as we shall see, the stability condition on the parameters and is considerably more complex. As before, we have an amplification polynomial of the form of (A.5), now with

where as before, . Because is again multilinear in the three cosines, its extrema can only occur at the eight corners of the cubic region defined by , and . These extrema are

The non-positivity requirement on then amounts to requiring that these extreme values be non-positive. The resulting stability region in the plane is shown in grey in Figure A.10(a).

Assuming that and fall in this region, we must now find the values of which satisfy (A.9). The minimum value of depends on and in a non-trivial way; referring to Figure A.10(a), the stability domain can be divided into three regions, and in each there is a different closed form expression for the upper bound on . These bounds are given explicitly in the caption to Figure A.10(a).

In order to examine the directional dependence of the dispersion error, we may expand in a Taylor series about , as was done in the (2+1)D case. We have

which implies that

and the dispersion error is directionally-independent to fourth order. This special choice of the parameters and is plotted as a dotted line in Figure A.10(a). It is well worth comparing this optimization method with the computer-based techniques applied to the same problem in [158].

The computational and add densities for the scheme will be

Considerable computational savings are possible if any of , , or is zero.

Finally, we remark that the (3+1)D interpolated scheme can be realized as a waveguide mesh, where, at any given junction, we will have four types of waveguide connections: those of admittances , and are connected to the neighboring junctions located at gridpoints at distances , and away respectively, and a self-loop of admittance is also connected to every junction. We end up with exactly difference scheme (A.30), with

The passivity condition is then a positivity condition on these admittances, and thus on the parameters , , and . Recalling the expression for in terms of and from (A.31), we must have

This region is shown, in dark grey, in Figure A.10(b). The positivity condition on (expressed in terms of , and as per (A.31)) gives the bound on , which is

(for passivity) |