MDWD Networks as Multi-step Schemes

where is the frequency domain transform variable corresponding to any MD-causal coordinate , and is the frequency domain unit shift in the same direction.

For spatially inhomogeneous problems, this spectral mapping is equivalent to the application of the trapezoid rule in direction . We can thus write, using operator notation,

where is a shift operator defined by

when applied to any continuous function . Consider again the lossless (1+1)D transmission line equations

which can be written as

under the application of coordinate transformation (3.19) and using scaled variables and , as well as the scaled time variable . Under the substitution of (3.73), for , and using the generalized trapezoid rule in time, defined by

where is a shift in the scaled time direction of duration , we get

to second order in . This can be rewritten as

where we have used , , the fact that , and also the definitions of the port resistances of the MDWD network of Figure 3.14(b), given in (3.64)

with

The recursion corresponding to (3.77b) is very similar, under the interchange of and . Note that if and are constants, and if the difference scheme is operating at the CFL bound (so that , then (3.78) can be simplified to

which is a simple centered difference approximation to (3.74a) and which we will see again in the waveguide mesh context in §4.3.2. Unlike the case of the mesh however, away from the passivity bound we have a

In order to generate a scheme which operates on alternating interleaved grids (called *offset sampling* in [61]), it is possible to use a doubled time step of
in order to implement the generalized trapezoid rule applied to the time derivatives in (3.75a) and (3.75b), i.e.,

in which case we get, as an approximation to (3.74a),

where and are defined as per (3.79), but where is now equal to . This form also reduces to simple centered differences when and are constant, and when we are operating at the CFL bound.

The computational stencils corresponding to the two different schemes are shown in Figure 3.20; the top black dot in either picture represents the location of the grid variable currently being updated (either or ), and the other dots cover the discrete region of influence of the difference scheme. Notice in particular that each scheme has a width of only three grid points, corresponding to nearest-neighbor-only updating. Also, because these are multi-step methods, one might expect that we will have to take special care when initializing the scheme; we discuss this issue in §3.10. For the offset scheme of Figure 3.20(b), the stencil can be shifted one step to the left or right without any overlapping; thus such a scheme can subdivided into two mutually exclusive subschemes (operating only for always even or always odd), one of which may be dropped from the calculating scheme entirely. This behavior appears in many of the difference schemes which we will come across subsequently; we will pay particular attention to such schemes during a spectral analysis of finite difference schemes in Appendix A.

Finally, we note that in general, the determination of stability for a multi-step scheme can be quite difficult; even in the constant coefficient case, it will in general be necessary to perform *Von Neumann analysis* [176] (see Appendix A for such an analysis applied to difference schemes for the wave equation in (2+1)D and (3+1)D), which can be quite formidable. Here, however, we are ensured stability through the passivity condition on the network.