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

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Computing u^{\prime}

u is often displacement/position, u^{\prime} is velocity and can be computed by u^{\prime}(t_n) \approx \frac{u^{n+1}-u^{n-1}}{2\Delta t} = [D_{2t}u]^n

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