Passivity vs. Stability

Perhaps the single most interesting question resulting from this thesis can be simply stated: For a given stable finite difference scheme, where ``stable'' is to be taken in the sense of Von Neumann (see Appendix A), is there always a concretely passive network realization? Because the difference scheme coefficients and network element values are usually parametrized by $v_{0}$, the space step/time step ratio, the question is often one of the range of values of $v_{0}$ for which a given scheme is stable or passive. As we have seen, a distinction between passivity and stability manifests itself in various ways in many very different settings. We saw, for example, in Section 4.3.6, that several different waveguide networks for the transmission line problem, though all equivalent in infinite-precision arithmetic to the same simple centered difference approximation, are passive over quite different ranges of $v_{0}$, depending on material parameter variation. Even more striking examples will be seen in Appendix A, in the case of the triangular scheme for the (2+1)D wave equation, and in particular for so-called ``interpolated'' difference schemes for the wave equation in (2+1)D and (3+1)D; these are rudimentary constant-coefficient difference schemes, and yet the difference between the stability condition and the passivity condition for the equivalent waveguide mesh is already quite complex; for the other mesh structures examined in Appendix A, Von Neumann stability and passivity imply one-another. Other instances appear throughout this work. The question is one of network topology--that is, there are many network topologies corresponding to a given difference scheme, and though the stability bound on $v_{0}$ will be the same for all of them, the bounds for passivity will be, in general, distinct (see §A.2.3 and §A.2.4 for some interesting examples). In sum, passivity is a sufficient, but not necessary condition for numerical stability; it may well be, however, that it is always possible to find a particular topology such that these conditions imply one-another. This author would like very much to make sense of the ``grey area'' between the two conditions.