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Resulting scheme: $$ \begin{align} [D_t u]^{n-\half} &= v^{n-\half}, \tag{50} \\ [D_t v]^n &= \frac{1}{m}( F(t_n) - \beta |v^{n-\half}|v^{n+\half} - s(u^n))\tp \tag{51} \end{align} $$
The truncation error in each equation is found from $$ \begin{align*} [D_t \uex]^{n-\half} &= \vex(t_{n-\half}) + R_u^{n-\half},\\ [D_t \vex]^n &= \frac{1}{m}( F(t_n) - \beta |\vex(t_{n-\half})|\vex(t_{n+\half}) - s(u^n)) + R_v^n\tp \end{align*} $$ Using (3)-(4) for derivatives and (21)-(22) for the geometric mean: $$ \uex'(t_{n-\half}) + \frac{1}{24}\uex'''(t_{n-\half})\Delta t^2 + \Oof{\Delta t^4} = \vex(t_{n-\half}) + R_u^{n-\half},$$ and $$ \vex'(t_n) = \frac{1}{m}( F(t_n) - \beta |\vex(t_n)|\vex(t_n) + \Oof{\Delta t^2} - s(u^n)) + R_v^n\tp $$
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