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In this section, we summarize the basics of Von Neumann analysis
provided in [8]. Consider the (+1)D realvalued grid
function
, defined for integer and for
, the set of all integer
tuples. Such a grid function will be used, in a finite difference
scheme, as an approximation to the continuous solution
to some problem, at the location
, and at
time , where is the grid spacing, and is the time
step. Here, and henceforth in this appendix, we have assumed that the
grid spacing is uniform in all the spatial coordinates, and that the
spatial domain is unbounded. We define the space step/time step ratio
to be
The spatial Fourier transform of
is defined by
and is a periodic function of
, a vector of spatial wavenumbers. The transform can be inverted by
where
refers to the space enclosed by the intervals
, for
.
If, for a given grid spacing , we define the discrete spatial norm of
by
and the corresponding spectral norm of
by
then if
and
are in their respective spaces, Parseval's relation gives
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