The Mid-Point Method

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:

$$\Delta t f\left(x_n, t_n\right)$$ k₁ =

$$\Delta t f\left(x_n + \frac{1}{2} k_1, t_n + \frac{1}{2}\Delta t\right)$$ k₂ =

$$x_n + k_2 + {\cal O}\left((\Delta t)^3\right)$$ x_n+1 =

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

$$x_{n+1} = x_n + f\left(x_n + f\left(x_n, t_n\right) \frac{1}{2}\Delta t, t_n + \frac{1}{2}\Delta t\right)\Delta t$$ (4.77)

In other words, in this scheme we make a simple Euler step to the mid-point of the interval $[t_n, t_n + \Delta t]$, find the corresponding x_{n + 1/2} and evaluate $f\left(x_{n + 1/2}, t_{n+1/2}\right)$, and then use that value in order to jump from t_n to t_n+1.

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.