### 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

for all which are second-order differentiable with respect to and . On the right-hand side, we have introduced the following shift operator notation:
 (D.4)

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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