Next |
Prev |
Up |
Top
|
Index |
JOS Index |
JOS Pubs |
JOS Home |
Search
The finite difference scheme corresponding to a rectilinear mesh is
obtained by applying centered differences to the wave equation, over a
rectangular grid with indices
and
(which refer to points with
spatial coordinates
and
). The difference
scheme
is (see Equation (4.53) of [2])
 |
(14) |
and the amplification polynomial equation is of the form (5), with
for
. From (7), we thus have
and we have
Condition (8) is thus satisfied, and condition (9) gives the bound
which implies that the amplification factor
for such values
of
. Because
, this bound is the same
as the bound for passivity of the associated mesh scheme, given in
(4.63) of [2].
The amplification factors, however, are
distinct at all spatial frequencies only for
. If
, then the factors are degenerate for
, and for
and we are then in the situation discussed in §2.2
where linear growth of the solution may occur. This is an important
special case, because it corresponds to the standard finite difference
scheme for the rectilinear waveguide mesh (i.e. the realization
without self-loops). The waveguide mesh implementation does not allow
such growth at these frequencies2.
As far as assessing the computational requirements of the finite difference scheme, first consider the case
. Five adds are required at each grid point in order to update. Given that
, we can write the computational and add densities for the scheme as
for |
|
For
, however, scheme (13) simplifies to
 |
(15) |
which may be operated on alternating grids, i.e.,
need only be calculated for
even (or odd). The computational and add densities, for
are then
for |
|
where we note that the reduced scheme (14) requires only four adds for updating at a given grid point; in addition, the multiplies by
may be accomplished, in a fixed-point implementation, by simple bit-shifting operations. The increased efficiency of this scheme must be weighed against the danger of instability, and the fact that because grid density is reduced, the scheme is now applicable over a smaller range of spatial frequencies. The numerical phase velocities of the schemes, at the stability limit, and away from it, at
, are plotted in Figure 1. It is interesting to note that away from the stability limit, the numerical dispersion is somewhat less directionally dependent; this important factor may be useful from the point of view of frequency-warping techniques [4] which may be used to reduce numerical dispersion effects for schemes which are relatively directionally-independent. This idea has been discussed in the waveguide mesh context (where self-loops will be present) in [7].
Next |
Prev |
Up |
Top
|
Index |
JOS Index |
JOS Pubs |
JOS Home |
Search
Download vonn.pdf