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Consider the following one-step explicit difference scheme, which relates values of the grid function
to values at the previous time step:
where
is some subset of
, and the parameters
are constants; it is initialized by setting
equal to some function
(assumed to be in ). Taking the spatial Fourier transform of this recursion gives
so defined is called the spectral amplification factor for a one-step finite difference scheme. (2) implies that we have, in particular, that
|
(4) |
where
is the spatial Fourier transform of the initial condition
. (3) further implies that
and finally, through Parseval's relation, that
If the
which define the difference scheme are independent of the grid spacing and the time step, then such a difference scheme is called stable if
The norm of the solution to the difference equation will thus not increase as the simulation progresses.
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