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
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.,  for more examples.
Well Posed Initial-Value Problem
Cylinder with Conical Cap