Forward-Euler (FE)

The backward difference was based on the usual left-sided limit in the definition of the time derivative:

$$\dot x(t) \;=\;\lim_{\delta\to 0} \frac{x(t) - x(t-\delta)}{\delta} \;\approx\; \frac{x_n-x_{n-1}}{T}$$

The forward difference comes from the right-sided limit:

$$\dot x(t) \;=\;\lim_{\delta\to 0} \frac{x(t+\delta) - x(t)}{\delta} \;\approx\; \zbox{\frac{x_{n+1}-x_n}{T}}$$

• As $T\to0$ , the forward and backward difference approximations approach the same limit, because $x(t)$ is assumed continuous and differentiable at $t$

• The forward difference gives an explicit finite difference scheme for the force-driven-mass problem above, even if the driving force $f_n$ depends on current velocity $v_n$ :
  $$\hat{v}_{n+1} \;=\;\hat{v}_n + \frac{T}{m} f_n, \quad n=0,1,2,\ldots$$
  with $v_0\mathrel{\stackrel{\scriptscriptstyle\mathrm{\Delta}}{\scriptstyle=}}0$

• We obtain the same finite-difference scheme by introducing an ad hoc delay in the driving force of the Backward Euler scheme to get $\hat{v}_n = \hat{v}_{n-1} +(T/m)f_{n-1}$