The truncation error for quadratic damping (2)

For simplicity, remove the absolute value. The product becomes $$[D_t\uex D_t\uex]^n = (\uex'(t_n))^2 + \frac{1}{12}\uex(t_n)\uex'''(t_n)\Delta t^2 + \Oof{\Delta t^4}\tp$$

With $$m[D_t D_t\uex]^n = m\uex''(t_n) + \frac{m}{12}\uex''''(t_n)\Delta t^2 +\Oof{\Delta t^4},$$ and using $mu'' + \beta (u')^2 + s(u)=F$, we end up with $$R^n = (\frac{m}{12}\uex''''(t_n) + \frac{\beta}{12}\uex(t_n)\uex'''(t_n)) \Delta t^2 + \Oof{\Delta t^4}\tp$$ Second-order accuracy! Thanks to

• difference approximation with error $\Oof{\Delta t^2}$
• geometric mean approximation with error $\Oof{\Delta t^2}$