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Although the choice of the free parameter
which gives a maximally direction-independent numerical dispersion profile has been made, in the past, through computerized optimization procedures [157], we note here that it is possible to make a theoretical choice as well, based on a Taylor series expansion of the spectrum.
The spectral amplification factors for the interpolated scheme can be written in terms of the function
, or, equivalently, in terms of the function
. It should be clear, then, that if
is directionally-independent, then so are the amplification factors, and thus the numerical phase velocity (see §A.1.4) as well. Ideally, we would like
to be a function of the spectral radius
![$ \Vert$](img525.png)
![$ \beta$](img517.png)
alone. Now examine the Taylor expansion of
about ![$ \beta$](img517.png)
:
The directionally-independent
term reflects the fact that the scheme is consistent with the wave equation; higher order terms in general show directional dependence. The choice of
, however, gives
and the directional dependence is confined to higher-order powers of
. Thus for this choice of
, the numerical scheme is maximally direction independent about spatial DC. Note that this value of
does fall within the required bounds for a passive waveguide mesh implementation. The value of
(for which the numerical dispersion profile is plotted in Figure A.2), which is very close to
, was chosen by visual inspection of dispersion profiles for various values of
.
Next: The Triangular Scheme
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Stefan Bilbao
2002-01-22