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Finite Difference Time Domain Scheme

Using centered finite difference approximations (FDA) for the second-order partial derivatives, we obtain a finite difference scheme for the ideal wave equation [30,18]:

$\displaystyle {\ddot y}(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$ (2)
$\displaystyle y''(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2}
\protect$ (3)

where $ T$ is the time sampling interval, and $ X$ is a spatial sampling interval.

Substituting the FDA into the wave equation, choosing $ X=cT$, where $ c\isdeftext \sqrt{K/\epsilon }$ is sound speed (normalized to $ c=1$ below), and sampling at times $ t=nT$ and positions $ x=mX$, we obtain the following explicit finite difference scheme for the string displacement:

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m)$ (4)

where the sampling intervals $ T$ and $ X$ have been normalized to 1. To initialize the recursion at time $ n=0$, past values are needed for all $ m$ (all points along the string) at time instants $ n=-1$ and $ n=-2$. Then the string position may be computed for all $ m$ by Eq. (4) for $ n=0,1,2,\ldots\,$. This has been called the FDTD or leapfrog finite difference scheme [9].

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Download wgfdtd.pdf

``On the Equivalence of the Digital Waveguide and Finite Difference Time Domain Schemes'', by Julius O. Smith III, version published at (in PDF and PostScript formats only).
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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