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The Cubic Rectilinear Scheme
This is the simplest scheme for the (3+1)D wave equation. The grid points, indexed by , and are located at coordinates
. The finite difference scheme is written as

(27) 
If the grid points are located at the corners of a cubic lattice, then updating the scheme requires access to the grid function at the six neighboring corners; see Figure 8(a).
The stability analysis is very similar to that of the (2+1)D rectilinear scheme, except that we now have a 3tuple of spatial frequencies,
. The amplification polynomial equation is again of the form of (5), with
and thus
Because
is multilinear in the cosines, it is simple to show that
and so, from (9),
(for Von Neumann stability) 

When
, the amplification factors become degenerate and linear growth of the solution may occur for
, and for
. The computational and add densities are
for
, and
at the stability limit
. At this limit, the scheme may, like the (2+1)D scheme, be divided into two mutually exclusive subschemes. See Figure 8(b) and (c) for plots of the numerical dispersion properties of the cubic rectilinear scheme.
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