As we mentioned early on, the greater goal of this thesis was to provide a unified and comprehensive treatment of numerical integration methods based on wave and scattering concepts; indeed, published results are dispersed across a wide variety of journals and fields, and assembling them has been somewhat of a challenge in itself. Although as we have seen, wave digital filtering methods and digital waveguide networks are two sides of the same coin (and a more apt metaphor might be that they are but two facets among many on a large, unexplored crystal), the research communities are more than a little isolated from one another. The author has fallen victim to this kind of parochialism as well--one huge regret we have is that we were not made aware of the TLM method earlier in this project, because of its similarity to DWNs, and the richness of the family of structures that has been proposed. In the spirit of Fettweis et al., we have tried to keep the scope as general as possible, treating physical systems that appear across a wide range of disciplines: MDWDFs are just as applicable to musical acoustics and plate vibration problems as the DWN is to electromagnetic field simulation. The unambiguously rosy part of the story, however, ends here.

An interesting remark appears in the preface to a recent book [33] on the transmission line matrix method (TLM). (It was mentioned in §4.1.1 that TLM structures, in that they are constructed from discrete transmission lines, are very similar to digital waveguide networks, though in the case of TLM, apparently no link with digital filtering structures has been made.) The author of this book makes a few comments in a eulogy to J. B. Johns, the originator of this method:

One immediately gets a sense of the partisan spirit that must have been predominant at the time of the method's inception. In particular, TLM was seen (and still is, judging from this book, which dates from 1998) as a competitor to FDTD; the same could be said for wave digital filtering approaches. While going through most of the scattering simulation literature, one is bound to feel uneasy at times about the short shrift given to finite difference methods, especially in the WDF arena, where they have been almost willed out of existence; with the notable exception of [131], there has been almost no attempt to view MDWD networks as finite difference schemes (which they are). This is unfortunate, because there is a wealth of well-developed and powerful theory surrounding difference methods which has been in place for many years^{}. It may be that the lack of commentary is due to the self-evident nature of the link--what could these scattering methods be *but* finite difference schemes? The real reason, perhaps, is that difference methods are seen by some as old, crude, and worse, not a physically motivated means of performing a simulation. This point of view, while prejudicial, is partly justified, but begs the question: what is special about scattering methods? A thesis would seem to be the right place to least ask (if not fully answer) this question. While this was not the major ``research'' goal of this project, it was the most significant source of motivation behind its undertaking, and every effort has been made to make clear the strengths and weaknesses of scattering methods.

With a clearer picture of the relationship between difference and scattering methods at hand, one may get the feeling that somewhere behind the scenes, Rube Goldberg has been busy at work. Indeed, some of the strong features of MDWDFs, according to Fettweis [47], such as local interconnectedness and parallelizeability are possessed by simple finite difference methods such as FDTD as well, and difference methods are undeniably easier on the programmer. Some DWN researchers are even moving away from wave implementations in favor of difference realizations [157]. In balance, however, these circuit-based scattering methods do offer a uniquely physical approach to numerical simulation, especially in the wave digital framework (though as we saw in §4.10, the relationship between MDWD networks and DWNs is now firmly established). Having access to a passivity condition offers the algorithm designer the most simple means imaginable of ensuring numerical stability for complex problems even in the presence of boundary conditions. These stability properties carry over in finite arithmetic as well; this is as sure an indicator as any of the essential correctness of a numerical simulation approach (circuit-based or otherwise) that pays close attention to physics.

Still, this author feels, more now than at the beginning of this project, that proponents of scattering methods have more to learn from straight-ahead finite difference practitioners than they realize (and perhaps more than vice-versa as well, though the balance is probably slight).