An open problem

System (B.21) is of the same form as (B.13) (though note that we have neglected to perform the variable scalings), but it is written in terms of entropy variables ${\bf z}$ as opposed to non-conservative variables ${\bf w}$. Applying coordinate transformation (3.18), we can get the system into the form

$$\frac{1}{2}\left({\bf L}_{\mathcal{U}}D_{1}{\bf z}+ D_{1}{\bf L}_... ... M}_{\mathcal{U}}D_{2}{\bf z}+ D_{2}{\bf M}_{\mathcal{U}}{\bf z}\right)= {\bf0}$$ (B.22)

where

$${\bf L}_{\mathcal{U}} = v_{0}{\bf P}_{\mathcal{U}}+{\bf A}_{\math... ...{1.0in}{\bf M}_{\mathcal{U}} = v_{0}{\bf P}_{\mathcal{U}}-{\bf A}_{\mathcal{U}}$$

Both ${\bf L}_{\mathcal{U}}$ and ${\bf M}_{\mathcal{U}}$ will be positive definite if $v_{0}$ is chosen sufficiently large. Though system (B.22) would appear to be in the correct form for an MDKC representation, there are certain difficulties.

First, the matrices ${\bf L}_{\mathcal{U}}$ and ${\bf M}_{\mathcal{U}}$, unlike ${\bf L}$ and ${\bf M}$ from (B.16), are not diagonal. Fettweis was able to take advantage of the fact that in terms of the variables ${\bf w}$, all inter-loop couplings are linear and shift-invariant (the coupling matrix ${\bf N}$ is a constant), so nonlinearities can be well-isolated. This is no longer the case here. It is of course possible to write down an MDKC corresponding to (B.22)--the entries of ${\bf L}_{\mathcal{U}}$ and ${\bf M}_{\mathcal{U}}$ become inductances directly. But positive definiteness of ${\bf L}_{\mathcal{U}}$ and ${\bf M}_{\mathcal{U}}$ does not imply that the off-diagonal entries will be positive, so our MDKC will not necessarily be a concretely passive representation.

We might attempt to avoid this by treating (B.22) as a simple series combination of two vector inductors, of inductances ${\bf L}_{\mathcal{U}}$ and ${\bf M}_{\mathcal{U}}$. In analogy with definitions of the coupled inductances in §2.3.7, it is certainly possible to define lossless nonlinear coupled inductances by

$${\bf v}_{1} = \frac{1}{2}\left({\bf L}_{\mathcal{U}}D_{1}{\bf z}+ D_{1}{\bf L}_{\mathcal{U}}{\bf z}\right)$$

$${\bf v}_{2} = \frac{1}{2}\left({\bf M}_{\mathcal{U}}D_{2}{\bf z}+ D_{2}{\bf M}_{\mathcal{U}}{\bf z}\right)$$

and the resulting MDKC is essentially identical to that of Figure 3.6, for the simple linear advection equation, except that the current is now ${\bf z}$. It is difficult to introduce wave variables, however, because power-normalization is not straightforward in the nonlinear vector case; for an inductor of vector inductance ${\bf L}>{\bf0}$, the relationship analogous to (3.42) does not hold, i.e.,

$${\bf v} = \frac{1}{2}\left({\bf L}D_{j}{\bf i}+D_{j}({\bf Li})\right)\neq {\bf L}^{T/2}D_{j}\left({\bf L}^{1/2}{\bf i}\right)$$ (B.23)

The second form of the inductor (which is distinct, and also lossless) on the right of (B.23), involving some left square root ${\bf L}^{T/2}$ of ${\bf L}$, is that which would be essential for power-normalization, because then we would be able to write

$${\bf L}^{-T/2}{\bf v} = D_{j}({\bf L}^{1/2}{\bf i})$$

and then define power-normalized vector wave variables by

$$\underline{{\bf a}} = \frac{1}{2}\left({\bf R}^{-T/2}{\bf v}+{\bf R}^{1/2}{\bf i}\right)$$

$$\underline{{\bf b}} = \frac{1}{2}\left({\bf R}^{-T/2}{\bf v}-{\bf R}^{1/2}{\bf i}\right)$$

where ${\bf R}^{T/2}$ is the left square root of some positive definite matrix port resistance ${\bf R}$. Making the usual choice of ${\bf R} = 2{\bf L}/T_{j}$ (or rather ${\bf R}^{T/2} = \sqrt{2/T_{j}}{\bf L}^{T/2}$), we would then arrive at the familiar wave relationship of (3.38) in terms of the vector waves $\underline{{\bf b}}$ and $\underline{{\bf a}}$. The two inductor definitions of (B.23) do, however, coincide if the nonlinearity is confined to the diagonal elements of ${\bf L}$. This is precisely what Fettweis has taken advantage of in his formulation.

It would be of fundamental interest to know whether a passive MDKC for general nonlinear systems of the form of (B.22) (and its analogues in higher dimensions), amenable to wave digital discretization in fact exists. In such an MDKC or MDWD network, the global conserved quantity would have the interpretation of an entropy, which can be thought of as a generalized form of energy [68]. We also note that the numerical methods examined here can also be applied to fluid dynamic systems in (2+1)D [16] and (3+1)D [49]. The nonlinear algebraic systems to be solved become larger, but are still localized. Also, we mention that these numerical methods do not seem to reduce to conventional finite difference schemes along the lines of Godunov's method and its offspring [171].