Comment on Numerical Instability

We have just shown, in the derivation ending with (4.30), that scattering in a particular waveguide network can be rewritten as a finite difference scheme purely in terms of the junction quantities. Thus all numerical solutions obtained using the waveguide network implementation could also be obtained (at least in infinite-precision arithmetic) using such a scheme. It is interesting to note that certain solutions to the finite difference equation (4.30) can not be obtained using the DWN, if we require that the wave variables in the network be bounded in magnitude. As a very simple example, consider initializing scheme (4.24) with $U_{i}(0) = -1$ and $U_{i}(1) = 1$, for all $i$. Then we will have $U_{i}(2) = 3$, $U_{i}(3) = 5$, and in general, $U_{i}(n) = 2n-1$ for all $i$. Similar linear growth will result from setting $U_{i}(0) = U_{i}(1) = (-1)^{i}$. We will then have, at any future time step $n$, $U_{i}(n) = (2n-1)(-1)^{i+n-1}$.

Though these solutions would appear to be completely unphysical, it is worth mentioning that the (1+1)D wave equation (to which (4.24) is an approximation) admits linear growth as well; $u=t$, for example, is a solution to (4.18). It is possible to view this solution as the sum of two traveling wave solutions $(x/\gamma+t)/2$ and $(-x/\gamma + t)/2$; these, however are unbounded in magnitude, and thus the wave variables used to initialize the DWN will be as well; the finite difference scheme, on the other hand, produces this behavior for the bounded initial conditions mentioned above. It is important to note that this linear growth occurs at the spatial DC and Nyquist frequencies; it is simple to show that these are in fact the only spatial frequencies for which scheme (4.24) will admit such behavior. We will return to this point in some detail in Appendix A, because the analysis is somewhat easier in the frequency domain.