Runge-Kutta integration

The approximation used in (8.21) is known as Euler integration. It is the simplest approximation, but does not perform well enough in many practical settings. One of the most common improvements is the fourth-order Runge-Kutta integration method, which expresses the new state as

$$x_i[k+1] \approx x_i[k] + \frac{\Delta t}{6} (w_1 + 2 w_2 + 2 w_3 + w_4) ,$$ (8.23)

in which

$$\begin{split}w_1 & = f({\dot x}_i(k \Delta t)) w_2 & = f({... ...ot x}_i(k \Delta t + \Delta t) + \Delta t \; w_3) . \end{split}$$ (8.24)

Although this is more expensive than Euler integration, the improved accuracy is usually worthwhile in practice. Many other methods exist, with varying performance depending on the particular ways in which ${\dot x}$ is expressed and varies over time [134].