= Solution
Use the stage solution continuous from $k_2=f(y)$ as $h\to0$. Its displacement is $\delta=h[(1-a)f+ak_2]=hf+ah^2f'f+O(h^3)$. Taylor expansion of $f(y+\delta)$ gives
$$
k_2=f+hf'f+h^2\left(af'^2f+\tfrac12f''f^2\right)+O(h^3).
$$
Consequently the <Runge-Kutta method> advances by
$$
y_{n+1}=y+hf+\tfrac12h^2f'f+h^3\left(\tfrac a2f'^2f+\tfrac14f''f^2\right)+O(h^4).
$$
The exact solution has third-order term $h^3(f'^2f+f''f^2)/6$. The first two orders agree for every real $a$, but the $f''f^2$ coefficient is always $1/4$ instead of $1/6$. Therefore \b[the general nonlinear method has $\boxed{\text{order }2}$ for every fixed real $a$]. The special value $a=1/3$ matches the linear test equation at third order but does not raise the nonlinear order. This is the <two-stage Runge-Kutta family with an implicit second stage>.
Back to article page
