Finite Difference Approximation (FDA)

Consider the simple differential equation relating velocity and force for an ideal mass:

$$\begin{eqnarray*} f(t)=m\frac{dv}{dt} \end{eqnarray*}$$

Finite Difference Approximation:

$$\begin{eqnarray*} \frac{dv}{dt} &\approx& \frac{v_n-v_{n-1}}{T} \qquad\quad\hb... ...{v_{n+1}-v_{n-1}}{2T} \qquad\hbox{(\lq\lq centered difference'')}\\ \end{eqnarray*}$$

E.g.,

$$v_n=v_{n-1} + \frac{T}{m} f_n, \quad n=0,1,2,\ldots\,.$$

(FDA for a force-driven mass)