Consistency
A finite-difference scheme is said to be
consistent with the original
partial differential equation if, given any sufficiently
differentiable function , the differential equation operating
on
approaches the value of the finite difference equation
operating on
, as
and
approach zero.
Thus, in the ideal string example, to show the consistency of Eq.(D.3)
we must show that
![$\displaystyle \left(\frac{\partial^2}{\partial x^2}
- \frac{1}{c^2}
\frac{\par...
...eft[
(\delta_x + \delta_x^{-1})
-
(\delta_t + \delta_t^{-1})
\right] y_{n,m}
$](http://www.dsprelated.com/josimages_new/pasp/img4454.png)



In particular, we have

In taking the limit as and
approach zero, we must maintain
the relationship
, and we must scale the FDS by
in
order to achieve an exact result:

as required. Thus, the FDS is consistent. See, e.g., [481] for more examples.
In summary, consistency of a finite-difference scheme means that, in the limit as the sampling intervals approach zero, the original PDE is obtained from the FDS.
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