One of the remedies to the problem that we have with
the Euler method is to sample more frequently.
Consider, for example, the following scheme:

k_{1} |
= | ||

k_{2} |
= | ||

x_{n+1} |
= |

We can combine all those three equations together to obtain the following expression:

(4.77) |

In other words, in this scheme we make a simple Euler step to the mid-point of the interval , find the corresponding

The mid-point method is second-order accurate.

The mid-point method is the simplest example of the family
of Runge-Kutta methods. It is also referred to as
the *second-order* Runge-Kutta method.