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 [484,314]:

$${\ddot y}(t,x) \approx \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$$ (E.1)

$$y''(t,x) \approx \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$$ (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:

$$y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m)$$ (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 [128].