# A clarification on “Runge–Kutta–Fehlberg” method

numerical methods

To apply the RK45 method to a system of ODEs represented as:
$$\dot{\overline{z}}=\overline{F}$$

At each time step you should first compute

\begin{align*} R_1 &= h\cdot \overline{F}(t_n,\overline{z}_n)\\ R_2&= h\cdot \overline{F}\left(t_n+\frac{1}{4}h,\overline{z}_n+ \frac{1}{4}R_1\right)\\ R_3&=h\cdot \overline{F}\left(t_n+\frac{3}{8}h,\overline{z}_n+ \frac{3}{32}R_1+ \frac{9}{32}R_2\right)\\ R_4&=h\cdot \overline{F}\left(t_n+\frac{12}{13}h,\overline{z}_n+\frac{1932}{2197}R_1-\frac{7200}{2197}R_2+\frac{7296}{2197}R_3\right)\\ R_5&=h\cdot \overline{F}\left(t_n+h,\overline{z}_n+\frac{439}{216}R_1-8R_2+\frac{3680}{513}R_3-\frac{845}{4104}R_4\right)\\ R_6&=h\cdot \overline{F}\left(t_n+\frac{1}{2}h,\overline{z}_n-\frac{8}{27}R_1+2R_2-\frac{3544}{2565}R_3+\frac{1859}{4104}R_4-\frac{11}{40}R_5\right) \end{align*}

Then you have:
\begin{align*} \overline{z^1}_{n+1}&=\overline{z}_n+\frac{25}{216}R_1+\frac{1408}{2565}R_3+\frac{2197}{4104}R_4-\frac{1}{5}R_5\\ &\\ \overline{z^2}_{n+1}&=\overline{z}_n+\frac{16}{135}R_1+\frac{6656}{12825}R_3+\frac{28561}{56430}R_4-\frac{9}{50}R_5+\frac{2}{55}R_6 \end{align*}

and my question is, to determine the error of this particular time step is it just the $$\left \| \overline{z^1}_{n+1}-\overline{z^2}_{n+1} \right \|$$ (norm of the difference between the two) or do you have a different formular for that?

Moreover, I would really appreciate if you can provide a suitable reference which contain this information. I have read several papers and it is a bit cumbersome….