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
for all
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., [484] 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.