A method of numerically integrating ordinary differential equations by using a trial step at the midpoint of an interval to
cancel out lower-order error terms. The second-order formula is

(1)

(2)

(3)

(where
is a Landau symbol), sometimes known as RK2, and
the fourth-order formula is

(4)

(5)

(6)

(7)

(8)

(Press et al. 1992), sometimes known as RK4. This method is reasonably simple and robust and is a good general candidate for numerical solution of differential
equations when combined with an intelligent adaptive step-size routine.