The Finite Difference Approximation

In the musical acoustics literature, the normal method for creating a
computational model from a differential equation is to apply the
so-called *finite difference approximation* (FDA) in which
differentiation is replaced by a finite difference (see Appendix D)
[484,314]. For example

and

where is the time sampling interval to be used in the simulation, and is a spatial sampling interval. These approximations can be seen as arising directly from the definitions of the partial derivatives with respect to and . The approximations become exact in the limit as and approach zero. To avoid a delay error, the second-order finite-differences are defined with a compensating time shift:

The odd-order derivative approximations suffer a half-sample delay error while all even order cases can be compensated as above.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University