A generalization of singly-implicit Runge-Kutta methods Original Research Article

The concept of effective order allows for the possibility that the result computed in a Runge-Kutta step is an approximation to some quantity more general than the actual solution at a step point. This generalization is applied here to singly-implicit methods. The limitation that requires severe and inconvenient restrictions on the abscissae in the method is removed and all that is now needed is that the abscissae be distinct. Finally, the construction of several low order methods will be detailed.