### 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

*shift operator notation*:

In particular, we have

*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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