Explicit Runge-Kutta Methods

Explicit Runge-Kutta Methods

Runge-Kutta methods To compute yn+1 only yn is needed. A series of k values is calculated which sample the derivatives of the states between t=tn and t=tn+1 as in the following example:

where h is the current step length, i.e. h=tn+1 -tn.

With this method only two k values are used and it is known as a 2-stage method. In a search for more accuracy, more k values are used. As the number of the stages increases, the order of the method tends to increase. It is not possible to define this term precisely here but very roughly speaking, the higher the order of the method, the greater the potential accuracy. In the most popular ERK methods 4, 5 or 6-stages are used giving orders of 4 and 5.

ERK methods are the easiest methods to implement (because they are explicit). Most modern algorithms implementing these formulae employ automatic step control. They can give good results provided the equations being solved are not too difficult. They are totally unsuitable for stiff problems.