Introductory Remarks

The process through which one arrives at a particular DWN intended to simulate the behavior of a distributed physical system has been, to date, quite different. Following the wave digital approach, one first obtains a multidimensional circuit representation (MDKC) of a system of PDEs, then applies a set of coordinate transformations and spectral mappings in order to obtain a discrete time/space algorithm. As discussed in the previous chapter, all MD circuit elements (as well as Kirchoff connections between elements) are to be interpreted as *distributed*, from the outset through to the final wave digital network. The integrity of each multidimensional circuit element (including its energetic properties) is preserved through the discretization step, as is network topology as a whole. As we mentioned in §1.1.2, however, the DWN is usually thought of as a collection of lumped elements, and as such, there has not as yet been a convenient multidimensional representation for such a network. We will address this point in some detail in the last section of this chapter. A DWN always operates on a predefined grid, at the points of which are located scattering junctions. Even though the paired delay elements (*waveguides*) which connect the various scattering junctions behave like transmission lines, we will persist in calling them lumped elements, because they are typically connected between junctions at neighboring grid points, and their behavior is hence localized in a way that that of a MDWD element is not.

A multidimensional WD network will behave consistently with the generating system of PDEs because the continuous-to-discrete spectral mapping applied approximates differential operators consistently; for a DWN, we must first show consistency of a DWN with a particular physical system. For both approaches, convergence of simulation results to the true solution of the physical system follows from this consistency as well as stability implied by passivity [176].

It was shown in [200] and [198] that the DWN structures designed to solve the *wave equation* in (2+1) and (3+1)D could be recast as *finite difference approximations* [176] (and in particular *centered difference approximations*) to these equations; we looked at the (2+1)D waveguide mesh briefly in §1.1.2. In infinite-precision arithmetic, these DWNs and centered differences yield identical results. A similar correspondence holds for the MDWD networks examined in the last chapter, though the equivalent difference methods are more involved (see §3.9). The distinction between a DWN and a finite difference approximation is in the types of signals used. Finite difference methods operate using grid variables which are approximations to the physical dependent variables of the problem at hand, but the DWN propagates wave variables; in this formulation, the solution to a system of PDEs is obtained as a by-product of the scattering of these waves. It is perhaps best to think of the difference between the finite difference scheme and DWN implementations as analogous to the distinction between direct form and lattice/ladder form digital filters [79]--both can be designed to implement the same transfer functions, but for the latter forms, stability is tightly controlled by the range of values which the filter multipliers (``reflection coefficients'') can take. And indeed, as we saw in §1.1.1, a particular type of (1+1)D DWN can be shown to be directly related to these lattice/ladder forms [165]. One goal of this chapter and the next is to show how this correspondence between the DWN and centered differences may be extended to a wide variety of physical systems.

The immediate question which arises is then: If the DWN is equivalent to finite differences, then is there a compelling reason for using it? Finite differences, after all, are more straightforward to implement. The answer is two-fold. First, although the approaches are equivalent in infinite precision arithmetic, this is no longer true when we are forced, inevitably, to truncate both the signals and multipliers in a computer implementation; stability of a DWN can be simply maintained even in finite arithmetic. Second, the stability criterion for a DWN is, as for MDWD networks, a *positivity condition* on the values of the elements contained in the network (i.e., the immittances of the transmission line segments). It thus becomes very simple to check stability of a given DWN, even in the presence of boundary conditions. Checking the stability of a finite difference scheme is considerably more involved, especially considering that a difference scheme which is stable over the interior of a domain may become unstable when boundary conditions are applied [82]. There is a theoretical machinery for performing such checks (known as GKSO theory [82,176]), though it can be formidable even in the (1+1)D case. It is quite possible, of course, to *design* a convergent numerical method using a DWN, and then to *apply* it as a finite difference scheme; as mentioned above, however, its stability in finite arithmetic is then no longer guaranteed.

A full technical summary of this chapter appeared in §1.3.