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]:

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

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

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