Study guide: Truncation Error Analysis

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The central difference for $u'(t)$ (1)

$u(t_{n+\half}) - u(t_{n-1/2}) = u'(t_n)\Delta t + \frac{1}{24}u'''(t_n) \Delta t^3 + \Oof{\Delta t^5} \tp$ By collecting terms in

$[D_t u]^n - u(t_n)$ we find

$R^n$ to be

$\begin{equation} R^n = \frac{1}{24}u'''(t_n)\Delta t^2 + \Oof{\Delta t^4}, \end{equation}$ Note:

Second-order accuracy since the leading term is $\Delta t^2$

Only even powers of $\Delta t$

The central difference for u′(t) u'(t) (1)

The central difference for $u'(t)$ (1)