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 L) [458,292]. For example

$${\dot y}(t,x)\approx \frac{y(t,x)-y(t-T,x)}{T}$$ (G.2)

and

$$y'(t,x)\approx \frac{y(t,x)-y(t,x-X)}{X}$$ (G.3)

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

$${\ddot y}(t,x) \approx \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$$ (G.4)

$$y''(t,x) \approx \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2}$$ (G.5)

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