# Integration of ODE¶

## Runge-Kutta¶

$\begin{split}z_0 &= y(x) \\ z_1 &= z_0 + h f(x,z_0) \\ z_{m+1} &= z_{m-1} + 2h f(x+mh,z_m) \\ y(x+H) &\approx y_n = \frac{1}{2} \left( z_n + z_{n-1} + h f(x+H,z_n) \right) .\end{split}$
This method contains only the even powers of $$h$$ thus we can gain two orders of precision at a time by calculating one more correction.