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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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Linear model without damping

\begin{equation} u''(t) + \omega^2 u(t) = 0,\quad u(0)=I,\ u'(0)=0\tp \tag{36} \end{equation} Centered difference approximation: \begin{equation} [D_tD_t u + \omega^2u=0]^n \tag{37} \tp \end{equation} Truncation error: \begin{equation} [D_tD_t \uex + \omega^2\uex =R]^n \tp \end{equation} Use (15)-(16) to expand [D_t D_t\uex]^n : [D_tD_t \uex]^n = \uex''(t_n) + \frac{1}{12}\uex''''(t_n)\Delta t^2, Collect terms: \uex''(t) + \omega^2\uex(t)=0 . Then, \begin{equation} R^n = \frac{1}{12}\uex''''(t_n)\Delta t^2 + \Oof{\Delta t^4} \tp \end{equation}

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