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Geometric mean: $$ \begin{align} [\overline{u^2}^{t,g}]^{n} &= u^{n-\half}u^{n+\half} = (u^n)^2 + R^n, \tag{21}\\ R^n &= - \frac{1}{4}u'(t_n)^2\Delta t^2 + \frac{1}{4}u(t_n)u''(t_n)\Delta t^2 + \Oof{\Delta t^4} \tp \tag{22} \end{align} $$
Harmonic mean: $$ \begin{align} [\overline{u}^{t,h}]^{n} &= u^n = \frac{2}{\frac{1}{u^{n-\half}} + \frac{1}{u^{n+\half}}} + R^{n+\half}, \tag{23}\\ R^n &= - \frac{u'(t_n)^2}{4u(t_n)}\Delta t^2 + \frac{1}{8}u''(t_n)\Delta t^2 \tp \tag{24} \end{align} $$
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