Evaluating the truncation error

To evaluate the effect of the truncation error, we consider the following sinusoidal function

\[ u(t)=U\sin\left( 2\pi\frac{t}{T} \right) = U\sin(\omega t), \quad \omega=\frac{2\pi}{T}, \]

whose first derivative is \(U\omega \cos(\omega t)\). In Fig. 2, the error of the first derivative is shown for the forward, backward, centred and 4th order difference formulas.

Fig. 2 Error of the first derivative formulas.

As \(\omega \Delta t \to 0\), the error of the 4th order formula decreases much faster than those of the centred (2nd order) and forward/backward (1st order) formulas.