Effective order of Partitioned Runge-Kutta methods

We have constructed partitioned Runge-Kutta methods of effective order 4 with 3 stages. The most obvious feature of these methods is efficiency in terms of implementation cost. The numerical results verify that the asymptotic error behavior of the effective order 4 partitioned Runge-Kutta methods with 3 stages is similar to that of classical 4th order methods which necessarily require 4 stages. Hence effective order methods are more efficient than their classical order counterpart.