This considerable increase in attainable order for a given number of stages does not necessarily make implicit Runge-Kutta methods more computationally efficient than explicit ones. In order to apply an R-stage implicit Runge-Kutta method, at each stage it is necessary to solve the system of implicit non-liner equations which define the . This needs to be done by some iterative process. The discussion on this would be dealt with later in these notes.

To overcome this implicitness, a somewhat less formidable problem arises in the case when , and the resulting methods are called Semi-explicit Runge-Kutta methods. .

One can find that R-stage semi-explicit method can attain higher order than an R-stage explicit Runge-Kutta method as demonstrated by the following fourth order three-stage method given by

(6.14)