In this paper, we study diagonally implicit iteration methods for solving implicit Runge-Kutta-
Nystr??m (RKN) methods on parallel computers. These iteration methods are such that in each step, the
iterated method can be regarded as a diagonally implicit Runge-Kutta-Nystr??m method (DIRKN method).
The number of stages of this DIRKN method depends on the number of iterations and may vary from step to
step. Since a large number of these stages can be computed in parallel, and since the total number of stages
can be kept small by a suitable choice of the parameters in the iteration process, the resulting variable-stage
DIRKN methods are efficient on parallel computers.
