The truncation error for quadratic damping (1)

Definition of $R^n$: $$\begin{equation} [mD_t D_t \uex]^n + \beta |[D_{t} \uex]^{n-\half}|[D_t \uex]^{n+\half} + s(\uex^n)-F^n = R^n\tp \end{equation}$$

Truncation error of the geometric mean, see (21)-(22), $$\begin{align*} |[D_{t} \uex]^{n-\half}|[D_t \uex]^{n+\half} &= [|D_t\uex|D_t\uex]^n - \frac{1}{4}u'(t_n)^2\Delta t^2 + \\ &\quad \frac{1}{4}u(t_n)u''(t_n)\Delta t^2 + \Oof{\Delta t^4}\tp \end{align*}$$ Using (3)-(4) for the $D_t\uex$ factors results in $$\begin{align*} [|D_t\uex|D_t\uex]^n &= |\uex' + \frac{1}{24}\uex'''(t_n)\Delta t^2 + \Oof{\Delta t^4}|\times\\ &\quad (\uex' + \frac{1}{24}\uex'''(t_n)\Delta t^2 + \Oof{\Delta t^4}) \end{align*}$$