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!