On the order of Runge Kutta methods reusing last stage

In this paper we consider explicit Runge--Kutta (RK) methods for the numerical solution of Initial Value Problems (IVPs) in differential equations in which the last function evaluation in a step is reused, substituting the first evaluation for the next step. This requirement implies that, except for the first step, the computational cost of an $s$--stage explicit RK method reduces from $s$ to $(s-1)$ function evaluations per step. It will be seen that, in general, if a RK method has order $p$, when introducing this reused stage the new method has in general order $<p$.
In this context we study the conditions on the coefficients of an explicit RK method with order $p$ such that the global error of the method reusing the last stage has the same order $p$ of accuracy. A detailed study on the conditions that the coefficients of an $s$ stage method with order $p$ must satisfy for the method reusing the last stage has also order $p$. In particular, for the families of $s$--stage explicit RK methods with the highest order $p=s$ we identify those methods reusing the last stage which retain the order of of the original method. The results of some numerical experiments are presented to confirm the theoretical results.