Replacing Backward-Difference by Forward-Difference

• Alternate finite difference scheme:
  $$\dot x(t) \eqsp \lim_{\delta\to 0} \frac{x(t+\delta) - x(t)}{\delta} \;\approx\; \frac{x[(n+1)T]-x(n T)}{T}$$

• As $T\to0$ , the forward and backward difference operators approach the same limit, since $x(t)$ is assumed continuous

• The forward difference gives an explicit finite difference scheme even if the driving force depends on current velocity:
  $$v_{n+1} \eqsp v_n + \frac{T}{m} f_n, \quad n=0,1,2,\ldots$$
  with $v_0\isdeftext 0$ .