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The schemes examined so far have all been spatially accurate to second-order. That is, at any time step, the norm of the difference between the numerical solution and the solution to the model problem will be proportional to
. In this section, we examine a family of explicit two-step schemes which are fourth-order spatially accurate. This family is more computationally intensive, due to the fact that updating the grid function requires access to past values which are two grid points away; in addition, we will see that a passive waveguide mesh implementation will not be possible in this case. These disadvantages are mitigated by the fact that the numerical dispersion is greatly reduced, so that the use of a coarse grid may be possible.
A Fourth-order Scheme
This scheme is, like the standard rectilinear scheme, defined over a grid with indices and which refer to a location with coordinates and . Updating, in this case, at a given point, requires access to values of the grid function at the previous time step at the set of 25 grid points which are located at most away in either the or directions, as shown in Figure 7(a). The difference scheme will have the general form
In order for (24) to approximate the wave equation, we first require that the constants , , , , and satisfy the constraints
Then, to ensure that the scheme is fourth-order spatially accurate, we additionally require
We can then write all the parameters in terms of , and , as
These constraints are all arrived at through a tedious but
straightforward Taylor series expansion of the scheme. As for the
interpolated scheme discussed in §3.2, passivity is
guaranteed by a simple positivity condition on the scheme parameters,
in this case
. From (27c), it should be
clear that if and , then we must necessarily have
, and a passive waveguide mesh implementation for this
scheme is ruled out. This is not to say that fourth-order spatially
accurate DWNs do not exist; we showed, in  that such a
network does exist, at least in the case of the (1+1)D transmission
line system (the wave equation is a special case of this system). The
conclusion is that the topology of the form discussed in this section
does not permit a mesh realization, but there are other forms that do.
The amplification polynomial for this scheme is of the form of (5), with
In order to determine stability bounds, we are faced with finding the extrema of
in terms of the parameters and . Because
is not multilinear in the cosines, finding these extrema explicitly is a challenging problem.
Let us first simplify the class of difference schemes by looking for those which exhibit maximally direction-independent numerical dispersion. As in §3.2, we expand
in a Taylor series about
, to get
The absence of a term in
reflects the fourth-order accuracy of the scheme. If we choose
, however, we get
and the scheme is direction-independent to sixth order in .
Making use of this setting for in terms of ,
now depends only on the free parameter ; through a computer analysis, it is possible to show that condition (8) is satisfied for . The upper bound on , from condition (9) is plotted as a function of in Figure 6.
We have plotted a numerical dispersion profile in Figure 7(b). It is interesting to note that the maximum value of
for this family of schemes would always appear to be slightly greater than 1, although the numerical phase velocity does indeed approach the physical velocity at spatial DC (as it will for any consistent scheme).
The computational and add densities for this scheme are, in general,
There are several ways of cutting down on computational costs; for example, because and are free parameters, we may simply set them to zero, and the add density is significantly reduced. There is, however, no decomposition of this scheme into mutually exclusive subschemes.
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