Computing truncation errors in nonlinear problems (2)

With (3)-(4), (35) leads to $R^{n+\half}$ equal to $$\uex'(t_{n+\half}) + \frac{1}{24}\uex'''(t_{n+\half})\Delta t^2 - f(\uex^{n+\half},t_{n+\half}) - \frac{1}{8}\uex''(t_{n+\half})\Delta t^2 + \Oof{\Delta t^4} \tp$$ Since $\uex'(t_{n+\half}) - f(\uex^{n+\half},t_{n+\half})=0$, the truncation error becomes $$R^{n+\half} = (\frac{1}{24}\uex'''(t_{n+\half}) - \frac{1}{8}\uex''(t_{n+\half})) \Delta t^2\tp$$ The computational techniques worked well even for this nonlinear ODE!