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 A.7(a). The difference scheme will have the general form

In order for (A.24) to approximate the wave equation, we first require that the constants , , , , and satisfy the constraints

(A.26) |

Then, to ensure that the scheme is fourth-order spatially accurate, we additionally require

(A.27) |

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 §A.2.2, passivity is guaranteed by a simple positivity condition on the scheme parameters, in this case . From (A.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 §4.10.5 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 (A.5), with and

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 §A.2.2, 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 (A.8) is satisfied for . The upper bound on , from condition (A.9) is plotted as a function of in Figure A.6.

We have plotted a numerical dispersion profile in Figure A.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.