Runge-kutta methods

Runge-Kutta (RK) methods for numerical integration of ordinary differential equations are popular because of their simplicity and efficiency. They are considered a class different from the predictor-corrector methods or PC methods, although the RK and PC methods are very similar in many respects.

The fourth-order RK method of Nystrom for second-order equations

has proved very useful in application to the tracking of charged particles in magnetic fields. It consists of the formulae

This formulation is taken from Abramowitz74 (algorithm 25.5.20), and minimizes the number of evaluations of the magnetic field (in practice only two evaluations per step are needed, since y₄ is close to y(x+h), and the next step can be started using the latest field values from the present step).

If applied to the equation File:Hepa img970.gif , written in the form File:Hepa img971.gif with File:Hepa img972.gif , the Nystrom method reduces to Simpson's rule

For a more detailed discussion, e.g. Wong92, Press95, or Flowers95.