### 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

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.

**Next Section:**

Well Posed Initial-Value Problem

**Previous Section:**

Cylinder with Conical Cap