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\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\vex}{{v_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)}

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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:

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