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 as opposed to non-conservative variables . Applying coordinate transformation (3.18), we can get the system into the form
First, the matrices and , unlike and from (B.16), are not diagonal. Fettweis was able to take advantage of the fact that in terms of the variables , all inter-loop couplings are linear and shift-invariant (the coupling matrix 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 and become inductances directly. But positive definiteness of and 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
. In analogy with definitions of the coupled inductances in §2.3.7, it is certainly possible to define lossless nonlinear coupled inductances by
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 . We also note that the numerical methods examined here can also be applied to fluid dynamic systems in (2+1)D  and (3+1)D . 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 .