... time²

$u = t$, for instance, satisfies (1).

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... equation²

Regrettably, a full discussion of consistency of difference schemes would take us too far afield, and we refer to [8] for a full exposition. The idea, grossly speaking, is that for a stable difference scheme, consistency is our guarantee that the numerical solution to the difference scheme converges to the solution of the continuous model problem as the grid spacing and time step are decreased. It is usually checked via a Taylor expansion of the difference scheme.

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... frequencies²

As an example of such growth at the spatial DC frequency, consider initializing the scheme (14) using $U_{i,j}(0) = 1$ for $i+j$ even and $U_{i,j}(1) = -1$ for $i+j$ odd. Then we will have $U_{i,j}(n) = 2n-1$, for $i+j+n$ even. It is simple to show that a waveguide implementation does not allow us to choose bounded wave variable initial conditions which yield these values for $U_{i,j}(0)$ and $U_{i,j}(1)$.

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... open²

We consider this to be the single most important issue raised in this thesis.

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