Although Euler integration is efficient and easy to understand, it generally yields poor approximations. Taking a Taylor series expansion of at yields
Runge-Kutta methods are based on using higher order terms of the Taylor series expansion. One of the most widely used and efficient numerical integration methods is the fourth-order Runge-Kutta method. It is simple to implement and yields good numerical behavior in most applications. Also, it is generally recommended over Euler integration. The technique can be derived by performing a Taylor series expansion at . This state itself is estimated in the approximation process.
The fourth-order Runge-Kutta integration method is
The approximation error depends on how quickly higher order derivatives of vary over time. This can be expressed using the remaining terms of the Taylor series. In practice, it may be advantageous to adapt over successive iterations of Runge-Kutta integration. In , for example, it is suggested that is scaled by , in which , the Euclidean distance in .
Steven M LaValle 2012-04-20