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.(N.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:

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., [495] 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.