We now make a few comments regarding the spectral properties of these difference methods; a detailed summary of spectral methods is provided in Appendix A.
Consider again the type II DWN for the (1+1)D transmission line equations, as discussed in §4.3.6. In the lossless, source-free case, the difference scheme can be written purely in terms of the junction voltages, and for integer time steps as
If the material parameters are constant, then (4.45) can be rewritten as
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In Figure 4.16, the quantity
is plotted for various values of the parameter
. At the stability bound, for
(i.e.
), both schemes are dispersionless. For the DWN, all spatial frequencies are slowed increasingly as
is decreased, but for the MDWD network, wave speeds decrease for
, then are exact again at
, and finally faster than the physical speed if
. This curious behavior of the phase velocities in the MDWD network was also mentioned in §3.9.2. In general, the phase velocities of the MDWD network are closer to the correct wave speed over the entire spatial frequency spectrum for a wide range of
--this is mitigated, however, by the fact this MDWD network corresponds to a three-step difference method (compared to two-step for the DWN), and is thus more computationally intensive.
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