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## MDKC and MDWD Network

The circuit can be derived along the same lines as for the (1+1)D case; we deal here with the discretization on a rectilinear grid, and will thus apply coordinate transformation defined by the of (3.22). Rewriting system (3.67) in terms of the new coordinates using , with the pseudo inverse (3.23) gives

where we have used the new current-like variables

and is, as in the (1+1)D case, an arbitrary positive constant (which has also been used to scale (3.69c)). will be treated as a simple time derivative, according to the generalized trapezoid rule discussed in §3.5.1. Figure 3.17 shows the MDKC that results from the transformed set of equations (3.69). The MDWD network corresponding to the MDKC is shown in Figure 3.18, where we have used step-sizes , .

Passivity follows from a positivity condition on the network inductances, in particular , and (the values of which are given in Figure 3.17). These conditions are

 (3.66)

The choice of , where and gives a stability bound of

 (3.67)

which is the best possible bound for this network [61]. Note that is again bounded away from the maximum group velocity, even taking into account the scaling factor ( in this case), which is a consistent feature of explicit numerical methods in multiple spatial dimensions.

If and are constant, and in addition , , , and are zero, and (3.71) holds with equality, i.e., we have

 (3.68)

then the network of Figure 3.18 simplifies to the structure shown in Figure 3.19. This particular structure bears a very strong resemblance to the (2+1)D waveguide mesh [157,198] which we saw briefly in §1.1.2, and will examine in detail in Chapter 4.

Next: Finite Difference Interpretation Up: The (2+1)D Parallel-plate System Previous: Phase and Group Velocity
Stefan Bilbao 2002-01-22