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Finite Difference Time Domain (FDTD) Scheme

As discussed in §C.2, we may use centered finite difference approximations (FDA) for the second-order partial derivatives in the wave equation to obtain a finite difference scheme for numerically integrating the ideal wave equation [481,311]:

$\displaystyle {\ddot y}(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$ (E.1)
$\displaystyle y''(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$ (E.2)

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:

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m)$ (E.3)

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.$ \,$(E.3) for $ n=0,1,2,\ldots\,$. This has been called the FDTD or leapfrog finite difference scheme [127].

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