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Consistency

A finite-difference scheme is said to be consistent with the original partial differential equation if, given any sufficiently differentiable function $ y(t,x)$ , the differential equation operating on $ y(t,x)$ approaches the value of the finite difference equation operating on $ y(nT,mX)$ , as $ T$ and $ X$ approach zero.

Thus, in the ideal string example, to show the consistency of Eq.$ \,$ (D.3) we must show that

$\displaystyle \left(\frac{\partial^2}{\partial x^2}
- \frac{1}{c^2}
\frac{\partial^2}{\partial t^2} \right)y(t,x)
= \lim_{T,X\to0}
\left[
(\delta_x + \delta_x^{-1})
-
(\delta_t + \delta_t^{-1})
\right] y_{n,m}
$

for all $ y(t,x)$ which are second-order differentiable with respect to $ t$ and $ x$ . On the right-hand side, we have introduced the following shift operator notation:
$\displaystyle \delta_t^k y_{n,m}$ $\displaystyle \isdef$ $\displaystyle y_{n-k,m}$  
$\displaystyle \delta_x^k y_{n,m}$ $\displaystyle \isdef$ $\displaystyle y_{n,m-k}
\protect$ (D.4)

In particular, we have

\begin{eqnarray*}
\delta_t y_{n,m}&\isdef & y_{n-1,m}\\
\delta_t^{-1} y_{n,m}&\isdef & y_{n+1,m}\\
\delta_x y_{n,m}&\isdef & y_{n,m-1}\\
\delta_x^{-1} y_{n,m}&\isdef & y_{n,m+1}.
\end{eqnarray*}

In taking the limit as $ T$ and $ X$ approach zero, we must maintain the relationship $ X=cT$ , and we must scale the FDS by $ 1/X^2$ in order to achieve an exact result:

\begin{eqnarray*}
\lefteqn{\lim_{T,X\to0}
\frac{1}{X^2}
\left[
(\delta_x + \delta_x^{-1})
-
(\delta_t + \delta_t^{-1})
\right] y_{n,m}} \qquad\qquad& &\\
&=&
\lim_{T,X\to0}
\left[
\frac{\delta_x + 2 + \delta_x^{-1}}{X^2}
-
\frac{\delta_t + 2 + \delta_t^{-1}}{c^2T^2}
\right] y_{n,m}\\
&\isdef & \left(\frac{\partial^2}{\partial x^2}
- \frac{1}{c^2}
\frac{\partial^2}{\partial t^2} \right)y(t,x)
\end{eqnarray*}

as required. Thus, the FDS is consistent. See, e.g., [483] 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA